Physics Problems Q&A - Recent questions and answers in Physics Problems
http://physics.qandaexchange.com/?qa=qa/physics-problems
Powered by Question2AnswerAnswered: How to use symmetry?
http://physics.qandaexchange.com/?qa=3473/how-to-use-symmetry&show=3482#a3482
<p>Imagine that the three $4\Omega$ resistors are removed, leaving two separate circuits. </p>
<p>Now join EF with a conducting wire so that they are at the same potential. No current will flow from one circuit into the other, because there is no path for it to return along, and charge which flows cannot accumulate in either of the two circuits. </p>
<p>How does the potential at C compare with that at D? How does the potential at A compare with that at B? The PD across each battery is divided equally across the resistors in the same circuit. Relative to the potentials at E=F as 0V, the potentials at A, B will be 2V and the potentials at C, D will be 1V.</p>
<p>What effect will it have if you re-insert resistors between AB, CD and EF? Since AB, CD and EF are all at the same potential, it will not make any difference if we connect AB, CD with wires or re-insert the $4\Omega$ resistors. No current will flow along these connecting wires/resistors. </p>
<p>So the currents in the $6\Omega$ and $3\Omega$ resistors are the same whether or not the $4\Omega$ resistors are there.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3473/how-to-use-symmetry&show=3482#a3482Wed, 22 May 2019 16:46:45 +0000Answered: The photocurrent in the detector assuming all rays to be paraxial w.r.t. lens
http://physics.qandaexchange.com/?qa=3472/the-photocurrent-the-detector-assuming-rays-paraxial-lens&show=3481#a3481
<p>The mistake in your calculation is that you are assuming that the light which is focussed by the lens into a point $S'$ at $60cm$ to the right of the lens acts like a <strong>point source</strong> which radiates equally in all directions.</p>
<p>This assumption is not correct. All of the light from the lens converges onto $S'$ in a cone, and diverges again <strong>only to the right</strong> into another cone with the same angle. In the paraxial approximation we can assume this light is spread uniformly across the base of the cone.</p>
<p>When the cone of light which diverges from $S'$ reaches the detector its radius is half that of the aperture of the lens. This can be found from the geometry of similar triangles. So the area $A$ of the circular base of this cone at the detector is $\frac14$ of that of the lens, ie $A=1cm^2$. </p>
<p>The area of the detector is only $0.5cm^2$ and is completely covered by the cone. So only half the light which is collected by the lens subsequently falls onto the detector.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3472/the-photocurrent-the-detector-assuming-rays-paraxial-lens&show=3481#a3481Wed, 22 May 2019 15:46:34 +0000Answered: Current through a particular branch of RC circuits
http://physics.qandaexchange.com/?qa=3474/current-through-a-particular-branch-of-rc-circuits&show=3476#a3476
<p>Initially no current flows through the $600k\Omega$ resistor. It all flows onto the uncharged capacitor. After a long time the capacitor will be fully charged and the current through the $600k\Omega$ resistor is the same as that through the battery, which is $\frac{80}{3}\mu A$. </p>
<p>So the current through the resistor increases exponentially from $0$ to a limit of $$I_{\infty} = \frac{80}{3} \mu A$$ The rate of increase is set by the <strong>time constant</strong> of the parallel combination of resistor and capacitor, which is $\tau=CR$, where $R=600k\Omega$ and $C=2.5nF$. Such an increase can be written as $$I(t)=I_{\infty}(1-e^{-t/ \tau})$$ </p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3474/current-through-a-particular-branch-of-rc-circuits&show=3476#a3476Wed, 22 May 2019 11:06:56 +0000Answered: Velocity wrt COM
http://physics.qandaexchange.com/?qa=3465/velocity-wrt-com&show=3470#a3470
<p>Both spheres have their initial velocities reversed when they come into contact again. This is a consequence of both the <strong>Conservation of Momentum</strong> and the <strong>Conservation of Energy</strong>.</p>
<p>This is essentially the same as throwing a ball up into the air. If air resistance can be ignored, so that there is no loss of energy, it returns to the Earth with the same speed as it was launched. The Earth also returns to the ball with the same speed at which it initially moved away.</p>
<p>In this case also there is no loss of energy between the initial impulse and the final collision between the spheres. The final gravitational potential energy is the same as the initial potential energy, because the spheres return to their initial positions. So the final total kinetic energy will be the same as the initial total kinetic energy.</p>
<p>In the COM frame the momentum of the big sphere is always equal and opposite to that of the small sphere. (The centre of mass frame is the same as the centre of momentum frame.) Therefore if there are no external forces, so that momentum is conserved, then the speed of the small sphere is always in the same proportion $\frac{M}{m}$ to that of the big sphere, and the KE of the small sphere is also always in the same proportion.</p>
<p>When this consequence of the conservation of momentum is combined with the conservation of energy, it can be deduced that the speeds of both particles will be the same when they return to their initial positions. </p>
<p>If there was a loss of energy due to internal forces the two particles would not have their initial speeds when they returned to the initial position, but because the conservation of momentum still applies the ratio of their two speeds would remain at all times the same as initially, and they would return to the initial position at exactly the same time.</p>
<p>If the ratio of velocities varied, while the total energy was conserved, then one particle would return to the starting point before the other. This would mean that the position of the COM had changed, which cannot happen if there are no external forces.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3465/velocity-wrt-com&show=3470#a3470Thu, 16 May 2019 19:54:06 +0000Answered: Net energy loss in infinite series of capacitors
http://physics.qandaexchange.com/?qa=3460/net-energy-loss-in-infinite-series-of-capacitors&show=3469#a3469
<p>You have forgotten that there is work done by the battery after $t=0$, so that there is energy lost before S1 is opened and S2 is closed.</p>
<p>The work done by the battery to transfer charge $Q$ onto the 1st capacitor is $W=QV=\frac{Q^2}{C}$.</p>
<p>The final charges on the capacitors are $\frac12Q, \frac14Q, \frac{1}{16}Q, \frac{1}{64}Q, ...$ The total energy stored in all the capacitors at this time is $$E_{\infty}=\frac{Q^2}{2C}[(\frac12)^2+(\frac14)^2+(\frac{1}{16})^2+(\frac{1}{64})^2 +...]=\frac{Q^2}{6C}$$ The total loss of energy after closing S1 is $$W-E_{\infty}=\frac{5Q^2}{6C}=\frac56CV^2=\frac{20}{6}\pi\epsilon_0 RV^2$$</p>
<p>So I think option A should be the correct answer. The marking scheme appears to be wrong. </p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3460/net-energy-loss-in-infinite-series-of-capacitors&show=3469#a3469Thu, 16 May 2019 18:46:46 +0000Answered: Electric field for volume charge distributed slab
http://physics.qandaexchange.com/?qa=3455/electric-field-for-volume-charge-distributed-slab&show=3463#a3463
<p>Because the slab extends infinitely in the xy plane, the electric field lies only along the z direction. The differential form of Gauss' Law for such a one-dimensional electric field is $$\frac{dE}{dz}=\frac{\rho(z)}{\epsilon_0}$$ This can be integrated, using the boundary condition that the electric field must be zero at large distances from the slab, because it is electrically neutral.</p>
<hr>
<p>A more elementary way of solving the problem is to divide the slab into infinitesimally thin layers of thickness $dz$. The electric field of each layer is uniform, independent of distance from the layer, and is $dE=\frac{d\sigma}{2\epsilon}$ pointing away from the layer on each side for +ve surface charge density $d\sigma=\rho(z)dz$. The total electric field at any point inside or outside of the slab is found by superposition of fields from every such layer in the slab.</p>
<p><strong>Note</strong> that the total electric field at any point due to all layers which are closer to the centre plane $z=0$, is zero. This is because the charge density is anti-symmetric : for every layer of +ve area charge density on one side of $z=0$ there is a layer of -ve charge with the same magnitude of area density on the other side of $z=0$. The electric fields of these two layers cancel out for points which outside of the two layers, in the same way that the total electric field is zero outside of a parallel plate capacitor (if the plate dimensions are very much bigger than the distance from them). </p>
<p>From this observation you can see that the electric field outside of the slab is zero, because all layers in the slab are closer to the centre plane.</p>
<p>The simplest way of getting the field <strong>inside</strong> the slab is to apply Gauss' Law using a "pill box" Gaussian surface which has one face A of area S at the surface of the slab $z=a$ (where $E(a)=0$) and the other face B at distance $|z|<a$ from the centre plane. The other face(s) of the pill box are parallel to the z direction so the electric flux through them is zero. There is no electric flux through face A; the flux through face B is $E(z)S=\frac{q}{\epsilon_0}$ where by integration $$q=\int_a^z S\rho(z)dz$$ is the total charge inside the pill box. </p>
<p>See <a rel="nofollow" href="https://physics.stackexchange.com/q/196936">Electric field in a non-uniformly charged sheet</a>. An answer identical to mine is given in the duplicate question <a rel="nofollow" href="https://physics.stackexchange.com/q/197230">Finding the electric field of a NON uniform slab?</a></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3455/electric-field-for-volume-charge-distributed-slab&show=3463#a3463Wed, 15 May 2019 21:58:01 +0000Find the equation of wave
http://physics.qandaexchange.com/?qa=3461/find-the-equation-of-wave
<blockquote><p>The linear mass density of a nonuniform wire under constant tension decreases gradually along the wire so that an incident wave is transmitted without reflection. The wire is uniform for $-\infty < x < 0$. In this region, a transverse wave has the form $$y(x, t) = 0.003 \cos (25x — 50t)$$ where $y$ and $x$ are in meters and $t$ is in seconds. In the region $0 < x < 20$ the linéar mass density decreases gradually from $\mu$ to $\frac14 \mu$. For $20 < x < +\infty$ the linear mass density is $\frac14 \mu$. Then prove that for $x > 20$ the wave equation is $$ y(x,t) = 0.0042 \cos (12.5 x — 50 t)$$</p>
</blockquote>
<p><img src="https://i.imgur.com/FkvLXz0.jpg" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3461/find-the-equation-of-waveWed, 15 May 2019 02:23:13 +0000Answered: To solve the equations of motion of a simple Pendulum $\ddot \theta = -\dfrac gl \sin \theta$
http://physics.qandaexchange.com/?qa=3454/solve-equations-motion-simple-pendulum-theta-dfrac-theta%24&show=3459#a3459
<p>You left out a minus sign in the 2nd equation. It should be written $$\ddot \theta \approx -\frac{g}{l}\theta$$ Now you can see that $\theta=e^{\sqrt{\frac{g}{l}} t}$ is not a solution because $\ddot \theta =+\frac{g}{l}\theta$. Instead we need a solution of the form $\theta=\pm e^{i\sqrt{\frac{g}{l}}t}$. Then we get $$\ddot \theta = i^2\frac{g}{l}e^{i\sqrt{\frac{g}{l}}t}=-\frac{g}{l}\theta$$ </p>
<p>Solutions to $$\ddot \theta = \pm\frac{g}{l}\theta$$ have the form $\theta=e^{\pm \sqrt{\frac{g}{l}} t}$ when the sign on the right is +ve and $\theta=e^{\pm i\sqrt{\frac{g}{l}} t}$ when the sign on the right is -ve. The most general solution is a linear combination : $$\theta_{+}=Ae^{+\sqrt{\frac{g}{l}} t}+ Be^{-\sqrt{\frac{g}{l}}t}=C\cosh(\sqrt{\frac{g}{l}} t+\phi)$$ $$\theta_{-}=Ae^{i\sqrt{\frac{g}{l}}t}+Be^{-i\sqrt{\frac{g}{l}} t}=C\cos(\sqrt{\frac{g}{l}} t+\phi)$$ for suitable values of $C, \phi$.</p>
<hr>
<p>Your 2nd equation $$\ddot \theta \approx -\frac{g}{l}\theta$$ is linear and is called the <strong>Harmonic Equation</strong>. Solutions are simply linear combinations of exponentials or trigonometric functions.</p>
<p>Your 1st equation $$\ddot \theta \approx -\frac{g}{l}\sin\theta$$ is non-linear. Solutions cannot be expressed in terms of simple exponential and trigonometric functions. They can be expressed in terms of <strong>elliptic functions</strong> as shown in <a rel="nofollow" href="https://www.math24.net/nonlinear-pendulum/">https://www.math24.net/nonlinear-pendulum/</a>.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3454/solve-equations-motion-simple-pendulum-theta-dfrac-theta%24&show=3459#a3459Tue, 14 May 2019 01:48:52 +0000Answered: Thermal stress in a bar.
http://physics.qandaexchange.com/?qa=3450/thermal-stress-in-a-bar&show=3451#a3451
<p>Young's Modules is defined as,</p>
<p>$Y = \frac{\sigma}{\epsilon}$</p>
<p>where $\sigma$ is the strain defined as $\sigma = F/A$ and $\epsilon$ is the stress defined as $\epsilon = \frac{\Delta L}{L_0}$.</p>
<p>In this case we require the stress, thus re-arranging the first equation gives,</p>
<p>$\sigma = \epsilon Y$</p>
<p>After plugging in the definition of $\epsilon$ we then arrive at,</p>
<p>$\sigma = \frac{\Delta L}{L_0}Y$</p>
<p>Now as you already worked out, $\Delta L = L_0\alpha\Delta T$. Therefore after plugging this in, we arrive at the required result:</p>
<p>$\sigma = \frac{L_0\alpha\Delta T}{L_0}Y = \alpha\Delta T Y$</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3450/thermal-stress-in-a-bar&show=3451#a3451Tue, 30 Apr 2019 11:18:41 +0000Answered: Container filled with fluid move with a acceleration a
http://physics.qandaexchange.com/?qa=3446/container-filled-with-fluid-move-with-a-acceleration-a&show=3449#a3449
<p>Your calculation is correct so far but it is not complete.</p>
<p>In the vertical direction there is a gravitational field of strength $g$ acting downwards. This is equivalent to an acceleration upwards at rate $g$. The acceleration of the ball relative to the container is $a_y=3g$ upwards.</p>
<p>In the horizontal direction the ball accelerates to the right at $a_x$ <em>relative to the container</em>. This is different from the acceleration $a$ of the container itself, which is equivalent to a gravitational field acting to the left. So we can think of the container as stationary and acted upon by gravitational fields downwards and to the left.</p>
<p>Horizontally the ball must cover twice the distance as vertically but in the same time. Since $s \propto a$ and $a_y=3g$ then we must have $a_x=6g$.</p>
<p>The final step which you have missed is to relate $a_x$ (the acceleration of the ball relative to the container) to the acceleration $a$ of the container itself. As in the vertical direction we have $a_x=3a=6g$. Therefore $a=2g$. </p>
<p>The time for the ball to reach the edge of the container is $\sqrt{\frac{L}{3g}}$ as you calculated. So the options B, C are correct.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3446/container-filled-with-fluid-move-with-a-acceleration-a&show=3449#a3449Fri, 19 Apr 2019 05:54:57 +0000time required to pass electricity through electroplating bath
http://physics.qandaexchange.com/?qa=3447/time-required-pass-electricity-through-electroplating-bath
<p>The time required to pass 3600 Coulomb of electricity through an electroplating bath using a current of 200 mA is <strong><strong>__</strong></strong>_ hours</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3447/time-required-pass-electricity-through-electroplating-bathWed, 17 Apr 2019 03:09:40 +0000Answered: Increasing resistance with temperature.
http://physics.qandaexchange.com/?qa=3432/increasing-resistance-with-temperature&show=3445#a3445
<p>The power delivered to the bulb is $P=V^2/R$. Assuming there are no heat losses, the power delivered to the bulb is related to the temperature increase of the filament by $P=c\frac{dT}{dt}$ where $c$ is the heat capacity of the bulb and $T$ is the temperature increase. Following your suggestion I shall model the variation of resistance with temperature as $R=R_0e^{\alpha T}$ where temperature $T=0$ at time $t=0$. Then $$\frac{dt}{dT}=\frac{cR_0}{V^2}e^{\alpha T}$$ which can be integrated to get $$t=\frac{cR_0}{\alpha V^2}(e^{\alpha T}-1)$$ For small values of $T$ such that $\alpha T \ll 1$ we get the approximation $$t\approx \frac{cR_0}{V^2}T$$</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3432/increasing-resistance-with-temperature&show=3445#a3445Mon, 15 Apr 2019 17:28:00 +0000Answered: Minimum force required to rotate a lamina when there is friction
http://physics.qandaexchange.com/?qa=3433/minimum-force-required-to-rotate-lamina-when-there-friction&show=3444#a3444
<p>This is quite difficult to solve but the result is fairly simple. Possibly there is an easier solution but I do not see it.</p>
<p>The friction force between the lamina and the rough ground is spread uniformly over the lamina. The force per unit area is $\sigma=\frac{\mu W}{A}$ where $W, A$ are the weight and surface area of the lamina and $\mu$ is the coefficient of friction.</p>
<p>Using polar co-ordinates with the fixed vertex as origin, the element of area is $dA=rdrd\theta$. The force on the element is $dF=\sigma dA$. The moment about the origin of the friction force on this element is $dM=rdF$. </p>
<p>If $p$ is the closest distance from the origin of the side of the lamina which does not pass through the origin, then the maximum distance from the origin at polar angle $\theta$ is $p\sec\theta$. The total moment of friction about the vertex is $$M=\sigma \int \int_0^{p\sec\theta} r^2drd\theta=\frac13 \sigma p^3\int \sec^3\theta d\theta$$ $$=\frac16 \sigma p^3 [\sec\theta \tan\theta+\ln|\sec\theta+\tan\theta|]$$ </p>
<p>See <a rel="nofollow" href="https://en.wikipedia.org/wiki/Integral_of_secant_cubed">Integral of Secant Cubed</a>. The limits of integration are $\theta_1, \theta_2$.<br>
<img src="https://i.imgur.com/8vyYA3N.png" alt=""><br>
Suppose the triangle has vertices ABC with side lengths $a< b< c$. With the axis at vertex A we have $$p=b, \theta_1=0, \sec\theta_1=1, \tan\theta_1=0, \sec\theta_2=\frac{c}{b}, \tan\theta_2=\frac{a}{b}$$ $$M_A=\frac16 \sigma [abc+b^3\ln(\frac{a+c}{b})]$$ Likewise when vertex B is the axis $$M_B=\frac16 \sigma [abc+a^3\ln(\frac{b+c}{a})]$$ For vertex C $$p=ab/c, \sec\theta_1=a/p, \tan\theta_1=a^2/cp, \sec\theta_2=b/p, \tan\theta_2=b^2/cp$$ $$c^3M_C=\frac16 \sigma [b^3(abc+a^3\ln(\frac{b+c}{a})+a^3(abc+b^3\ln(\frac{a+c}{b}))]=a^3M_A+b^3M_B$$ </p>
<p>The moment of friction is balanced by the moment of the applied force. When the axis is at vertices A, B the applied force is minimum when this force is applied at B, A respectively and it is perpendicular to AB; the moments are then $M_A=cF_A, M_B=cF_B$. When the axis is at vertex C the applied force is minimum when applied at A perpendicular to AC; then $M_C=bF_C$. Therefore $$bc^2F_C=a^3F_A+b^3F_B$$</p>
<p>In the question $\theta$ equals angle A so $$F_C=F_A\tan\theta \sin^2\theta +F_B\cos^2\theta $$</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3433/minimum-force-required-to-rotate-lamina-when-there-friction&show=3444#a3444Fri, 12 Apr 2019 21:02:01 +0000Answered: Calculating equivalent capacitance.
http://physics.qandaexchange.com/?qa=3442/calculating-equivalent-capacitance&show=3443#a3443
<p>The middle diagram shows the correct equivalent configuration. </p>
<p>The diagram on the right assumes that the potential is the same at all points of the interface between the upper and lower dielectrics. If this is true then the interface can indeed be replaced by a conducting plate and the upper dielectric becomes a single capacitor, as suggested in the right-hand diagram. </p>
<p>However, the interface is a dielectric surface not a conducting surface, so it need not be at the same potential at all points adjacent to each of the two lower dielectrics. The lower surface of the upper dielectric can be split into two surfaces which can be at different potentials. This is recognised in the middle diagram. </p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3442/calculating-equivalent-capacitance&show=3443#a3443Fri, 12 Apr 2019 18:12:45 +0000Answered: Velocity of image of submerged object when liquid surface moves
http://physics.qandaexchange.com/?qa=3437/velocity-image-submerged-object-when-liquid-surface-moves&show=3440#a3440
<p>Suppose the observer is a fixed distance $H$ above the coin. Then the distance from the observer to the water surface is $H-h$ and the apparent distance of the coin below the surface is $\frac34 h$. So the apparent distance of the coin from the observer (measuring downwards) is $$y=(H-h)+\frac34h=H-\frac14h$$</p>
<p>The velocity of the water surface is $\frac{dh}{dt}=+8m/s$ (plus because $h$ is increasing) so the rate at which the image of the coin is moving downwards (in the direction of increasing y) is $$\frac{dy}{dt}=-\frac14\frac{dh}{dt}=-2m/s$$ The minus sign indicates that the image is moving upwards towards the observer, whereas y increases downwards.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3437/velocity-image-submerged-object-when-liquid-surface-moves&show=3440#a3440Mon, 08 Apr 2019 16:12:29 +0000Given the vector current density, determine the total current flowing outward through a circular band
http://physics.qandaexchange.com/?qa=3431/current-density-determine-current-flowing-outward-circular
<p>Given the vector current density, determine the total current flowing outward through a circular band.:</p>
<p><a rel="nofollow" href="https://i.stack.imgur.com/klQAo.png"><img src="https://i.stack.imgur.com/klQAo.png" alt="enter image description here"></a></p>
<p>The answer should be 518A. It comes something around 3255 A. Where is the mistake?</p>
<p><a rel="nofollow" href="https://i.stack.imgur.com/g1T1l.jpg"><img src="https://i.stack.imgur.com/g1T1l.jpg" alt="enter image description here"></a></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3431/current-density-determine-current-flowing-outward-circularFri, 29 Mar 2019 11:20:51 +0000Nucleon-Nucleon Scattering
http://physics.qandaexchange.com/?qa=3430/nucleon-nucleon-scattering
<p><img src="https://i.imgur.com/hZhT97F.png[/img]" alt=""><br>
<img src="https://i.imgur.com/rklrj0w.jpg[/img]" alt=""></p>
<p>I computed the orbital angular momentum classically, and then used quantum mechanics. But I am not convinced of what I got, because $L \alpha \sqrt{E}$ is a proportional equation that does not justify that when E < 20 MeV you get L ~ 0 (I do not see, numerically speaking, the difference between either plugging 20 MeV or 19MeV).</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3430/nucleon-nucleon-scatteringWed, 27 Mar 2019 12:38:33 +0000Answered: How to apply the gauss law when charge density is a function of not only $r$?
http://physics.qandaexchange.com/?qa=3416/how-apply-the-gauss-law-when-charge-density-function-not-only&show=3428#a3428
<p>In general the electric potential (or electric field) at point P outside of a charge distribution can be expanded as a power series known as a <strong>multipole expansion</strong> : $$V(z)=\frac{A}{r}+\frac{B}{r^2}+\frac{C}{r^3}+\frac{D}{r^4}+...$$ where the coefficients $A, B, C, D, ...$ depend on the spherical polar and azimuthal co-ordinates $\theta, \phi$ and $r$ is the distance of P from centre O of the charge distribution. See <a rel="nofollow" href="http://physics.qandaexchange.com/?qa=3393/electric-field-far-away-of-a-bunch-of-charges">Electric field far away from a bunch of charges</a>.</p>
<p>In this case the total charge on the sphere is zero : $$Q=\int_0^R \int_0^{\pi} \rho r^2\sin\theta d\theta dr=KR\int_0^R (R-2r) dr \int_0^{\pi}\sin^2\theta d\theta=0$$ because the integral wrt $r$ is zero. So the electric potential far from the centre of the sphere will not be proportional to $1/r$ as it is for a point charge : that is, $A=0$ here. </p>
<p>It is convenient to divide the sphere into rings such as A and B which are parallel to the xy plane, because such rings have uniform charge density. A typical ring has radius $y=r\sin\theta$ and each point on it is the same distance $s$ from the point of interest P which lies on the $z$ axis. The potential at P due to the ring is simply $V=kQ/s$ where $k$ is Coulomb's constant,</p>
<p><img src="https://i.imgur.com/rIu95cR.png" alt=""></p>
<p>Using the Cosine Rule for ring A $$s^2=z^2-2rz\cos\theta+r^2=z^2(1-x)$$ where $x=p(c-p), c=2\cos\theta, p=\frac{r}{z}$. The charge on each ring is $$dQ=2\pi y \rho r d\theta dr=2\pi\rho r^2\sin\theta d\theta dr$$ </p>
<p>Ring B has the same charge as ring A and is located symmetrically about O, the centre of the sphere. The potential at P due to both rings A and B, each with charge $dQ$, is $$dV=\frac{dQ}{4\pi \epsilon_0 }(\frac{1}{s_1}+\frac{1}{s_2})=\frac{KR(R-2r)}{2\epsilon_0 z}(\frac{1}{\sqrt{1- x}}+\frac{1}{\sqrt{1+ x}})\sin^2\theta d\theta dr$$ </p>
<p>The reciprocals can be expanded as power series : $$\frac{1}{\sqrt{1-x}}=1+\frac12x+\frac38x^2+\frac{5}{16}x^3+\frac{35}{128}x^4+\frac{63}{256}x^5+\frac{231}{1024}x^6+...$$ $$\frac{1}{\sqrt{1+x}}=1-\frac12x+\frac38x^2-\frac{5}{16}x^3+\frac{35}{128}x^4-\frac{63}{256}x^5+\frac{231}{1024}x^6+...$$ which have the half-sum $$\frac12 (\frac{1}{\sqrt{1-x}}+\frac{1}{\sqrt{1+x}})=1+\frac38x^2+\frac{35}{128}x^4++\frac{231}{1024}x^6+...$$ </p>
<p>The advantage of taking 2 rings together is that we have halved the number of terms in the series. This simplifies calculation. To avoid double-counting we must reduce the range of integration to $\theta=0 \to \pi/2$.</p>
<p>The words "far from the centre" suggest that we should assume $z\gg R$ and therefore we ignore all except the lowest power terms. Substituting from above $$x^2=p^2(c-p)^2\approx \frac{r^2}{z^2}\cos^2\theta$$ </p>
<p>The potential at P is approximately $$V\approx \frac{KR}{\epsilon_0 z} \int_0^{\pi/2} d\theta \int_0^R dr . (R-2r)(1+\frac38 \frac{r^2}{z^2} \cos^2 \theta ) \sin^2\theta$$ $$= \frac{KR}{\epsilon_0 z} \int_0^R (R-2r)\frac38 \frac{r^2}{z^2} dr \int_0^{\pi/2} \cos^2 \theta \sin^2\theta d\theta$$ $$= \frac{KR}{\epsilon_0 z} (-\frac{R^4}{16z^2} )(\frac{\pi}{16})$$ $$= -\frac{\pi KR^5}{256\epsilon_0 z^3}$$ The electric field at point P is $$E_z=-\frac{dV}{dz}=-\frac{3\pi KR^5}{256\epsilon_0 z^4}$$</p>
<p><strong>Notes :</strong> </p>
<ol>
<li><p>We should expect the potential to be -ve because the outer part of the sphere (which is closest to P) is negatively charged. </p>
</li>
<li><p>The dominant term in the charge distribution is not a <em>dipole</em> $V\propto 1/r^2$ but a <em>quadrupole</em> $V\propto 1/r^3$. We should expect to lose the dipole term because the centres of +ve and -ve charge coincide. </p>
</li>
<li><p>This result gives only the leading term in the <strong>multipole expansion</strong> and applies only for $z\gg R$. We could obtain more terms by retaining powers of $x^4, x^6...$ in the half-sum expansion and higher powers of $r$ when substituting for $x$. As more terms are retained the result becomes more accurate for smaller values of $z \gt R$.</p>
</li>
</ol>
Physics Problemshttp://physics.qandaexchange.com/?qa=3416/how-apply-the-gauss-law-when-charge-density-function-not-only&show=3428#a3428Tue, 26 Mar 2019 16:51:47 +0000Where do I make the approximations in this proton and electron collision problem?
http://physics.qandaexchange.com/?qa=3407/where-make-approximations-proton-electron-collision-problem
<blockquote><p>A proton moving with a velocity $\beta c$ collides with a stationary electron of mass $m$ and knocks it off at an angle $\theta$ with the incident direction. Show that the energy imparted to the electron is approximately $$T \approx \frac{2\beta^2 \cos^2\theta}{1-\beta^2\cos^2\theta}mc^2$$</p>
</blockquote>
<p><em>Link to question :</em><br>
<a rel="nofollow" href="https://www.chegg.com/homework-help/questions-and-answers/proton-moving-velocity-collides-stationary-electron-mass-m-knocks-angle-incident-direction-q35643739?trackid=VRP2b2Jt">https://www.chegg.com/homework-help/questions-and-answers/proton-moving-velocity-collides-stationary-electron-mass-m-knocks-angle-incident-direction-q35643739?trackid=VRP2b2Jt</a></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3407/where-make-approximations-proton-electron-collision-problemSun, 17 Mar 2019 04:34:00 +0000Answered: Maximum charge and maximum current
http://physics.qandaexchange.com/?qa=3406/maximum-charge-and-maximum-current&show=3414#a3414
<p>This problem can be solved by two methods : (A) using conservation of flux linkage and energy, and (B) using differential equations.</p>
<p>What you are missing in your solution is the conservation of energy. No energy is lost because there is no resistance in the circuit after the battery is disconnected. </p>
<p>(<strong>Note :</strong> With the type of switch depicted in the diagram, the circuit is broken - although only for a short time. The current through $L_1$ cannot then flow. This causes a large back emf and a spark across the switch, losing most of the energy stored in the inductor. To avoid loss of energy the switch needs to be smooth, with a small amount of overlap, so that there is never a break in the circuit containing the inductor.)</p>
<p><strong>How the Circuit Behaves</strong></p>
<p>While the switch is in position 1 there is a current $i_0=E/r$ flowing through $L_1$ and (presumably) none through $L_2$. It is possible that a sinusoidal current flows around circuit QRUT before the switch is thrown. However, we are not told anything about this so we must assume that there was initially no current in $L_2$.</p>
<p>When the switch is changed to position 2 the current $i_0$ continues to flow through $L_1$ - it cannot change instantaneously. For the same reason current cannot flow through $L_2$ immediately - it takes time to build up. Instead, all of the current from $L_1$ flows initially onto the capacitor. </p>
<p><img src="https://i.imgur.com/ZZgJrs6.png" alt=""></p>
<p>There is initially no voltage across either inductor or the capacitor. As charge $q$ builds up on the capacitor there is an increasing voltage across all 3 components. This voltage increases the (upward) current $i_2$ through $L_2$ and decreases the (downward) current $i_1$ through $L_1$. </p>
<p>When the charge (and voltage) on the capacitor reaches a maximum, no more current flows into the capacitor. At this instant the current through $L_2$ is the same as that through $L_1$ - ie $i_1=i_2$ - because no current flows into or out of circuit PQTS. You can find the (maximum) charge on the capacitor at this instant using conservation of energy.</p>
<p>Although the charge and voltage on the capacitor then both decrease, this voltage continues to increase the (upward) current in $L_2$ and to decrease the (downward) current in $L_1$ until the charge on the capacitor becomes zero again. The current through $L_2$ is then a maximum because the polarity of the capacitor then switches, the voltage on it reverses, and this reduces the (upward) current $i_2$ while also increasing downward current $i_1$. By applying the conservation of energy again you can find the maximum current through $L_2$.</p>
<p><strong>(A) Solution using Conservation of Energy</strong></p>
<p>When the switch is thrown the current through $L_1$ is $i_0=E/r$. (This notation differs from yours.) Flux linkage is conserved in loop PQTS because this loop contains only inductors. When the currents in $L_1, L_2$ are generally $i_1, i_2$ we can write $$L_1i_0=L_1i_1+L_2i_2$$ </p>
<p>When the charge on the capacitor is a maximum $q_0$ then $i_1=i_2$ so $(L_1+L_2)i_1=L_1i_0$. By conservation of energy we can write $$L_1i_0^2=L_1i_1^2+L_2i_2^2+\frac{1}{C}q^2=(L_1+L_2)i_1^2+\frac{1}{C}q_0^2=\frac{L_1^2}{L_1+L_2}i_0^2+\frac{1}{C}q_0^2$$ $$q_0^2=\frac{CL_1L_2}{L_1+L_2}i_0^2$$ As we shall find later we can write this in the form $$q_0=\frac{i_0}{\omega}$$ where $\omega$ is the angular frequency of oscillations in the circuit.</p>
<p>When the charge on the capacitor becomes zero again ($q=0$) then the current through $L_2$ reaches its maximum value $i_3$. Equations for the conservation of flux linkage and energy then become $$L_1i_0=L_1i_1+L_2i_3$$ $$L_1i_0^2=L_1i_1^2+L_2i_3^2=L_1i_0^2-2L_2i_0i_3+\frac{L_2^2}{L_1}i_3^2+L_2i_3^2$$ $$0=[(\frac{L_2}{L_1}+1)i_3-2i_0]L_2i_3$$ We have already met one solution $i_3=0$; the other solution is $$i_3=\frac{2L_1}{L_1+L_2}i_0$$ We can go on to find the <strong>minimum current</strong> through $L_1$ which is $$i_4=\frac{L_1-L_2}{L_1+L_2}i_0$$</p>
<p><strong>(B) Solution using Differential Equations</strong></p>
<p>The voltage across the components in the circuit are related to the currents and charges by $$V=-L_1\frac{di_1}{dt}=L_2\frac{di_2}{dt}=\frac{1}{C}q$$ $$i_1-i_2=\frac{dq}{dt}$$ From these we get the equations $$L_1i_1+L_2i_2=L_1i_0$$ $$\frac{d^2q}{dt^2}+\omega^2 q=0$$ $$\omega^2=\frac{1}{C}(\frac{1}{L_1}+\frac{1}{L_2})$$ with the solutions $$q=q_0\sin\omega t$$ $$(L_1+L_2)i_1=L_1i_0+L_2\frac{dq}{dt}=L_1i_0+L_2\omega q_0\cos\omega t$$ $$(L_1+L_2)i_2=L_1i_0-L_2\frac{dq}{dt}=L_1i_0-L_1\omega q_0\cos\omega t$$ Substitute $i_1=i_0$ when $t=0$ into the 1st eqn to get $$i_0=\omega q_0$$ We can then write general solutions for the currents as $$i_1=i_0\frac{L_1}{L_1+L_2}(1+\frac{L_2}{L_1}\cos\omega t)$$ $$i_2=i_0\frac{L_1}{L_1+L_2}(1-\cos\omega t)$$ The minimum current through $L_1$ and the maximum current through $L_2$ are respectively $$i_4=\frac{L_1-L_2}{L_1+L_2}i_0$$ $$i_3=\frac{2L_1}{L_1+L_2}i_0$$</p>
<p><img src="https://i.imgur.com/OBmwT1g.png" alt=""></p>
<p><strong>Notes :</strong></p>
<ol>
<li><p>The current $i_2$ through $L_2$ always has a minimum of zero and is always positive (upward in the diagram). </p>
</li>
<li><p>Unlike $i_2$, the current $i_1$ through $L_1$ <strong>can become negative</strong> (flow upwards in the diagram). This happens if $L_1 \lt L_2$.</p>
</li>
<li><p>The charge on the capacitor is a maximum (both polarities) when the same current flows through both inductors - ie $i_1=i_2$.</p>
</li>
<li><p>When this happens the current through both inductors equals half the maximum current through $L_2$ - ie $$i_1=i_2=\frac{L_1}{L_1+L_2}i_0=\frac12 i_3$$ </p>
</li>
</ol>
Physics Problemshttp://physics.qandaexchange.com/?qa=3406/maximum-charge-and-maximum-current&show=3414#a3414Sat, 16 Mar 2019 18:41:11 +0000Answered: Problem 5.13 Griffiths; Balancing magnetic attraction with electrical repulsion
http://physics.qandaexchange.com/?qa=3412/griffiths-balancing-magnetic-attraction-electrical-repulsion&show=3413#a3413
<p>The line charges must have some mass. Because of this it is impossible to accelerate them to the speed of light. </p>
<p>Increasing their speed also increases their inertia. Initially almost all of the work done increases their speed and almost none increases their inertia. As their speed approaches the speed of light, almost all of the work done on them increases their inertia and there is little increase in speed.</p>
<p>So their speed will always be less than the speed of light, although you can get as close to $c$ as you wish.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3412/griffiths-balancing-magnetic-attraction-electrical-repulsion&show=3413#a3413Thu, 14 Mar 2019 19:29:11 +0000Griffiths 5.4 Force on a square loop
http://physics.qandaexchange.com/?qa=3404/griffiths-5-4-force-on-a-square-loop
<p><img src="https://i.imgur.com/HckbPJf.png[/img]" alt=""></p>
<p><img src="https://i.imgur.com/9SEg3Li.png[/img]" alt=""></p>
<p>Before doing any calculation, I see the net force being zero;</p>
<p>$$ F_{AB} = -F_{CD}$$</p>
<p>$$ F_{BC} = -F_{DA}$$ </p>
<p>But apparently this is not the case. The provided solution is:</p>
<p><img src="https://i.imgur.com/75ijztI.png[/img]" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3404/griffiths-5-4-force-on-a-square-loopSat, 09 Mar 2019 11:53:06 +0000How to induce large current in flexible wire and explaining how does it stretch into a circle
http://physics.qandaexchange.com/?qa=3401/induce-large-current-flexible-explaining-does-stretch-circle
<blockquote><p>The idea is to explain how could we induce current in a flexible loop using an initially free-of-current wire.</p>
</blockquote>
<p><strong>Some background.</strong></p>
<p>The current has to be the same all the way around (otherwise charge would be gathered at some point and in that accumulation of charge, the electric field would point in a way that the flow would even out). </p>
<p>The two forces involved in driving the current around the loop are the source force (typically a battery) and the electrostatic force, which smooths out the flow. The line integral of the vector sum of the previous forces is known as electromotive force (emf).</p>
<p><strong>My answer to the problem</strong> </p>
<p>My difficulty in this problem is that I do not have clear the sketch. I would say that the following happens:</p>
<p>Imagine that the loop is in the xy plane. The velocity of the current is tangential to the loop and its direction is clockwise; $B$ will point in the $-z$ direction and the force will point radially outwards. This force would be the source one (provided by the battery and the one that does work to move the loop; I know magnetic force does no work; the individual radial vectors of the force do work but the net work done by all of them is zero) if the wire were just connected to a battery. I also understand that the net electrostatic force is zero (this is a closed path).</p>
<p>But here there is no battery, the current is induced by a change in the current of the wire</p>
<p>My intuition tells me that is the change in the current of the wire what changes the flux of the wire and the changing flux induces the emf in the loop. But not really sure of this...</p>
<p><strong>What's the nature of the induced current in the loop?</strong></p>
<p><strong>EDIT:</strong> The original exercise does not provide details on the set up, but my interpretation was the following: we start with two free-of-current wires: one is a loop, the other a straight line. We apply current on the straight wire, which induces an emf on the loop (which will stretch into a circle; this idea came from one of the sources I checked: <a rel="nofollow" href="https://physics.stackexchange.com/questions/239591/magnetic-potential-energy">https://physics.stackexchange.com/questions/239591/magnetic-potential-energy</a> ).</p>
<p>Souces:</p>
<p><a rel="nofollow" href="https://physics.stackexchange.com/questions/239591/magnetic-potential-energy">https://physics.stackexchange.com/questions/239591/magnetic-potential-energy</a></p>
<p><a rel="nofollow" href="https://physics.stackexchange.com/questions/266895/what-force-causes-the-induced-emf-of-a-loop-and-the-difference-between-a-loop?rq=1">https://physics.stackexchange.com/questions/266895/what-force-causes-the-induced-emf-of-a-loop-and-the-difference-between-a-loop?rq=1</a></p>
<p>Introduction to Electrodynamics by David Griffiths</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3401/induce-large-current-flexible-explaining-does-stretch-circleFri, 08 Mar 2019 16:08:33 +0000Answered: Questions from a jumping kangaroo
http://physics.qandaexchange.com/?qa=3263/questions-from-a-jumping-kangaroo&show=3400#a3400
<p>A general algorithm which works for all functions $f(x)$ will have to use a <strong>numerical method</strong> of solution - ie start with an initial approximation to the trajectory, estimate how close this comes to the optimal solution (the "error"), use this to make an adjustment to the solution, estimate the new "error," and continue the same cycle until the "error" is small enough for your purposes.</p>
<p><strong>Analytic solutions</strong> in the form of closed functions are possible only for particular shapes of $f(x)$ such as rectangles, circles, and ellipses.</p>
<p>If the object is convex and has a vertical plane of symmetry then you can use the method in my answer to <a rel="nofollow" href="https://physics.stackexchange.com/questions/288193/">Minimum speed required to clear rectangular object</a>. This involves finding points A, B on the same horizontal level, at which the projectile could graze the obstacle at an angle of $45^{\circ}$ to the horizontal. The distance AB enables you to find the required speed at these points, and the height of AB above the ground enables you to find the minimum speed with which the object should be launched.</p>
<p>This appears to be the method which you are using in your calculation.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3263/questions-from-a-jumping-kangaroo&show=3400#a3400Mon, 04 Mar 2019 16:03:59 +0000Answered: Force that one half of uniformly charged solid sphere exerts on other half - Griffith 2.43
http://physics.qandaexchange.com/?qa=3368/force-uniformly-charged-solid-sphere-exerts-other-griffith&show=3399#a3399
<p><img src="https://imgur.com/yKhoEy9" alt=""></p>
<blockquote><p>The problem asks to calculate the force on one part due to other. But the electric field I'm using to do this calculation is the result E of complete solid non-conducting sphere. <strong>So why are we considering E of the northern hemisphere to calculate force it experience from the southern hemisphere?</strong></p>
</blockquote>
<p>This is a very good question.</p>
<p>One way of finding the total electrostatic force on the N hemisphere is to calculate the vector force $F_{ij}=k q_i q_j / r_{ij}^2$ on every charge $q_i$ in the N hemisphere due to every charge $q_j$ in the S hemisphere, then take the vector sum of all forces $F_{ij}$ : </p>
<p>$$F_N=\sum_{i \in N} \sum_{j \in S} F_{ij}$$</p>
<p>This double sum or integral is very difficult to calculate, for two reasons : </p>
<ol>
<li><p>the general expression for the force $F_{ij}$ between every pair of 2 charges is a complicated function using the Cartesian co-ordinate system. even more so when using spherical co-ordinates ;</p>
</li>
<li><p>each of the 2 charges $q_i, q_j$ in every pair has 3 co-ordinates, so in total there will be a sum ranging over 6 possible co-ordinates. </p>
</li>
</ol>
<p>An alternative method of solution (used in Jorge Daniel's answer) is to sum the <strong>total</strong> electrical force $F_i$ on each charge $q_i$ in the N hemisphere due to <strong>all</strong> of the charges in <strong>both</strong> the S and N hemispheres : </p>
<p>$$F_N=\sum_{i \in N} F_i=\sum_{i \in N} \sum_{j \in N,S} F_{ij}$$ </p>
<p>Whereas $F_{ij}$ is a very complicated expression, the total force $F_i$ on each charge is very much simpler because it is known to be radial and depends only on the radial distance of the charge $q_i$ from the centre of the sphere. </p>
<p>As you rightly point out, this means that within the expression for $F_i$ we will include the force $F_{ij}$ on charge $q_i$ due to other charges $q_j$ which are <strong>also in the N hemisphere</strong>. </p>
<p>However, if charge $q_j$ is also within the N hemisphere ($j \in N$), then when we take the sum over all charges $q_i$ in the N hemisphere ($i \in N$), this sum will include not only $F_{ij}$ but also the equal and opposite force $F_{ji}$ acting on $q_j$ due to charge $q_i$. These 2 contributions to the total force on all particles in the N hemisphere will cancel out because $F_{ij}=-F_{ji}$. </p>
<p>On the other hand, if charge $q_j$ is in the S hemisphere ($j \in S$) then the force $F_{ji}$ will <strong>not</strong> be included in the first summation over $i \in N$.</p>
<p>See <a rel="nofollow" href="https://physics.stackexchange.com/questions/23071/">Find the net force the southern hemisphere of a uniformly charged sphere exerts</a>.</p>
<hr>
<p>To make the above explanation clearer, suppose we have a system of only 4 charges $q_i$ with $i=1 \to 4$. Between every pair of charges there is an electrical force $F_{ij}$ meaning the force that charge $j$ exerts on charge $i$. </p>
<p>Suppose that charges $i=1 , 2$ constitute one object (which we call "the N hemisphere") and charges $i=3, 4$ constitute a second object ("the S hemisphere"). Then the total force on the N hemisphere due to the S hemisphere is $$F_N=F_{13}+F_{14}+F_{23}+F_{24}$$ The total forces on charges $q_1, q_2$ due to <strong>all</strong> other charges in <strong>both</strong> hemispheres (ie summing over $j \in N, S$) are $$F_1=F_{12}+F_{13}+F_{14}$$ $$F_2=F_{21}+F_{23}+F_{24}$$ If we add the forces on charges $q_1$ and $q_2$ (summing over $i \in N$) we get $$F_N'=F_1+F_2=(F_{12}+F_{21})+F_{13}+F_{14}+F_{23}+F_{24}=F_N$$ because $F_{12}=-F_{21}$. </p>
<p>Thus both methods give the same answer.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3368/force-uniformly-charged-solid-sphere-exerts-other-griffith&show=3399#a3399Mon, 04 Mar 2019 15:02:39 +0000question about 4-velocity
http://physics.qandaexchange.com/?qa=3396/question-about-4-velocity
<p>A particle is moving in the x, y plane at a speed of v = 0.80 and it is travelling an angle of 60 degrees<br>
above the x-axis.</p>
<p>(a) Rotate the spatial coordinates so that v lies along the x-axis and then construct the components of the 4-velocity for the particle.</p>
<p>(b) Construct the remaining orthonormal basis vectors for the particle in the rotated frame, then rotate the spatial coordinates back to find the basis vectors in the original frame.</p>
<p>(c) Perform the same steps above but this time begin by rotating the spatial coordinates so that <br>
v lies along the y-axis. Do you get the same result for the basis vectors after rotating back to the original frame? Should you? Draw a picture with ~v, the original (x, y) frame, and the two rotated frames to explain what is happening.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3396/question-about-4-velocityWed, 27 Feb 2019 22:32:29 +0000Answered: Rotating ball inside rotating cylinder.
http://physics.qandaexchange.com/?qa=3359/rotating-ball-inside-rotating-cylinder&show=3395#a3395
<p>I also do not understand the textbook solution.</p>
<p>In an equilibrium situation the ball is effectively rolling down a plane inclined at angle $\theta$ while the plane is itself accelerating upwards along the plane. In the ground frame of reference the centre of mass of the ball is stationary, so the resultant force on it must be zero. However, there is a non-zero torque which causes rotational acceleration down the plane.</p>
<p>The torque acting on the ball is $mgr\sin\theta$ where $r$ is its radius. Its acceleration down the plane, relative to the point of contact, is $r\beta$ where $\beta$ is its angular acceleration. Since there is no slipping at the point of contact, $r\beta$ must also equal the acceleration of the point of contact up the plane, which is $R\alpha$ where $R>r$ is the radius of the cylinder. That is : $$r\beta=R\alpha$$ </p>
<p>The equation of motion for the angular acceleration of the ball is therefore $$mgr\sin\theta=I\beta=\frac25 mr^2 \frac{R}{r}\alpha$$ $$\sin\theta=\frac{2R\alpha}{5g}$$</p>
<p>If there is a finite coefficient of friction $\mu$ then we must also have that $$\tan\theta \le \mu$$</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3359/rotating-ball-inside-rotating-cylinder&show=3395#a3395Wed, 27 Feb 2019 00:00:45 +0000Answered: Electric field far away from a bunch of charges
http://physics.qandaexchange.com/?qa=3393/electric-field-far-away-from-a-bunch-of-charges&show=3394#a3394
<p>You have not defined the problem well. "A bunch of charges" is a vague description. Both answers could be correct, or neither, depending on the arrangement of charges.</p>
<p>The electric field outside of any general system of charges can be modelled as a <a rel="nofollow" href="http://farside.ph.utexas.edu/teaching/jk1/lectures/node32.html"><strong>multi-pole expansion</strong> </a> of the form $$E=k(\frac{q_0}{r^2}+\frac{q_1}{r^3}+\frac{q_2}{r^4}+...)$$ The coefficients $q_k$ depend on the polar and azimuthal angles $\theta, \phi$ and are related to the <strong>spherical harmonic functions</strong>. The 1st term represents a monopole with the resultant charge on the bunch. The 2nd and 3rd terms represent a dipole and quadrupole respectively.<br>
<img src="https://i.imgur.com/dOpRZgf.png" alt=""><br>
If the "bunch of charges" has an overall non-zero charge $q_0$ then far from the bunch the 1st term will dominate. It will resemble a point charge with an electric field which varies approximately as $1/r^2$.</p>
<p>However, if the bunch has an overall zero charge $q_0=0$ but the centres of +ve and -ve charge do not coincide then the 2nd term will dominate. The electric field far from the bunch is will resemble that of a dipole, which varies as $1/r^3$.</p>
<p>If the centres of +ve and -ve charge do coincide, then the 2nd term will also vanish ($q_1=0$) - for example, 4 charges at the corners of a square, alternating. This arrangement could have a non-zero 3rd term (a quadrupole, $q_2 \ne 0$), for which the electric field is proportional to $1/r^4$.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3393/electric-field-far-away-from-a-bunch-of-charges&show=3394#a3394Mon, 25 Feb 2019 02:07:43 +0000Answered: Electric field intensity at the centre of a solid hemisphere with uniform volume charge density
http://physics.qandaexchange.com/?qa=3360/electric-intensity-centre-hemisphere-uniform-volume-density&show=3392#a3392
<p>The question is contradictory : It says that the charge density is uniform (the same at every point in the hemisphere), but also that it depends on azimuthal co-ordinate $\phi$, with $\rho=3\phi$. The latter is a very unusual variation, because it has a discontinuity at the meridian $\phi=0, 2\pi$. Perhaps a polar variation is intended, with $\rho=3\theta$, which has no discontinuity.</p>
<p><strong>(a) Uniform Volume Charge Density $\rho=\text{const}$</strong></p>
<p>Divide the hemisphere into rings (hoops) of thickness $dr$ and width $rd\theta$ located at constant values of $r, \theta$ parallel to the xy plane, with the centre as origin. The radius of the ring is $y=r\sin\theta$, its volume is $dV=2\pi y r d\theta dr$ and its charge is $dq=\rho dV$. </p>
<p>The z-component of the electric field due to this ring at the origin is $dE_z=-kdq \cos\theta/r^2$. The total electric field due to all rings in the hemisphere is $$E_z=-\int dE_z=-k\pi \rho \int_0^a dr \int_0^{\pi/2} \sin2\theta d\theta =-\frac{\rho a}{4\epsilon_0} $$ <br>
The total charge on the hemisphere is $Q=\frac23 \pi \rho a^3$ so the electric field can be re-written as $$E_z=-\frac{3Q}{8\pi \epsilon_0 a^2}$$</p>
<p><strong>(b) Polar Charge Density $\rho=3\theta$</strong></p>
<p>Again we can use rings because they have a constant value of $\theta$.</p>
<p>$$E_z=-\int 3\theta dV=-k\pi \int_0^a dr \int_0^{\pi/2} 3\theta \sin2\theta d\theta =-\frac{a}{4\epsilon_0} \frac34 [\sin2\theta-2\theta\cos2\theta]_0^{\pi/2} =-\frac{3\pi a}{16\epsilon_0}$$ </p>
<p>The total charge on the hemisphere is $$Q=\int dq=\pi\int_0^a 3r^2 dr \int_0^{\pi/2} \theta\sin2\theta.d\theta=\frac14 \pi^2 a^3$$ so the electric field can be re-written as $$E_z=-\frac{3Q}{4\pi \epsilon_0 a^2}$$</p>
<p><strong>(c) Azimuthal Charge Density $\rho=3\phi$</strong></p>
<p>We can no longer use rings because $\phi$ is not constant for a ring. The volume element is $dV=r^2\sin\theta dr d\theta d\phi$ and the charge on each element is $dq=3\phi dV$. </p>
<p>The electric field at the centre due to an element of charge at ($r,\theta,\phi$) has components $$dE_z=-k\frac{dq}{r^2}\cos\theta, dE_y=-k\frac{dq}{r^2}\sin\theta\cos\phi, E_x=-k\frac{dq}{r^2}\sin\theta\sin\phi$$ so we get $$E_z=-3k\int_0^a dr \int_0^{\pi/2} \frac12 \sin2\theta d\theta \int_0^{2\pi} \phi d\phi=-\frac{3\pi a}{2\epsilon_0}$$ $$E_y=-3k\int_0^a dr \int_0^{\pi/2} \frac12 (1-\cos2\theta) d\theta \int_0^{2\pi} \phi \cos\phi d\phi=-\frac{3 a}{2\epsilon_0}(\frac{\pi}{4}+1)$$ $$E_x=-3k\int_0^a dr \int_0^{\pi/2} \frac12 (1-\cos2\theta) d\theta \int_0^{2\pi} \phi \sin\phi d\phi=-\frac{3a}{2\epsilon_0}(\frac{\pi}{4}+1)$$<br>
The total charge on the hemisphere is $$Q=\int dq=\int_0^a 3r^2 dr \int_0^{\pi/2} \sin\theta d\theta \int_0^{2\pi} \phi d\phi=2\pi^2 a^3$$ so we can express the components of the field as $$E_z=-\frac{3Q}{4\pi \epsilon_0 a^2}, E_y=E_x=-\frac{3Q}{4\pi \epsilon_0 a^2}(\frac14+\frac{1}{\pi})$$</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3360/electric-intensity-centre-hemisphere-uniform-volume-density&show=3392#a3392Sat, 23 Feb 2019 23:06:15 +0000Answered: What is the fraction of work per second by F is converted into heat.
http://physics.qandaexchange.com/?qa=3373/what-is-the-fraction-of-work-per-second-by-converted-into-heat&show=3391#a3391
<p>Your equation #1 is $$(R+2\lambda x)i=B\ell v=B\ell \frac{dx}{dt}$$ Solving for $x$ as a function of time, and assuming $x(0)=0$, we get $$R+2\lambda x(t)=R e^{kt}$$ where $k=\frac{2\lambda i}{B\ell}$. The velocity of the bar is $v(t)=v_0e^{kt}$ where the initial velocity is $v_0=\frac{iR}{B\ell}$.</p>
<p>We do not need to find an expression for $F$. And we do not need to take account of a resistive force $Bi\ell$ which the magnetic field exerts on the wire. This is because the motion of the wire in the magnetic field is the cause of the current. The magnetic field does not create a current then exert a force on the same current which it has created. If the current had been generated independently of the magnetic field, <em>then</em> there would be a magnetic force on that current.</p>
<p>No work is done by the magnetic field. Only force $F$ does work. The rate of work $P$ done by force $F$ is the sum of 3 components : <br>
(1) the rate $K$ at which the kinetic energy of the bar is increasing;<br>
(2) the power $Q$ dissipated as heat in the resistors; and<br>
(3) the rate $M$ at which energy is being stored in the magnetic field created by the rectangular loop of current. </p>
<p>Now $M$ depends on the self-inductance $L(x)$ of the rectangular loop, which depends on $x$. Even for such a simple geometry an expression for $L(x)$ is difficult to obtain - see <a rel="nofollow" href="https://www.allaboutcircuits.com/tools/rectangle-loop-inductance-calculator/">Rectangle Loop Inductance Calculator</a> in All About Circuits website. I shall assume without justification that $M \ll P$ is negligible.</p>
<p>$$K=\frac{d}{dt}(\frac12mv^2)=mv\frac{dv}{dt}=mkv^2=mkv_0^2 e^{2kt}$$ $$Q=i^2(R+2\lambda x)=i^2Re^{kt}$$ </p>
<p>The fraction of work done by $F$ which is converted into heat is $$\frac{Q}{P}=\frac{Q}{Q+K}=\frac{1}{1+K/Q}=\frac{1}{1+he^{kt}}$$ where $$h=\frac{mkv_0^2}{i^2 R}=\frac{2\lambda m i R}{(B\ell)^3}$$ This is not constant, it decreases with time. The initial fraction at $t=0$ is $\frac{1}{1+h}$. If the rails have zero resistance ($\lambda=0$) then $k=K=h=0$ so $Q=P$ - ie all of the work done by $F$ is dissipated as heat in resistor $R$, as obtained in <a rel="nofollow" href="http://pleclair.ua.edu/ph102/Exams/Sum_2008/ph102_sum08_exam2_SOLN.pdf">Q4 in this worksheet</a>.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3373/what-is-the-fraction-of-work-per-second-by-converted-into-heat&show=3391#a3391Fri, 22 Feb 2019 17:48:17 +0000Answered: Find minimum distance between dipoles
http://physics.qandaexchange.com/?qa=3372/find-minimum-distance-between-dipoles&show=3390#a3390
<p>Your error is near the start of the calculation. The distance between diagonally opposite charged masses is $\sqrt{\ell^2+d^2}$ not $\sqrt{\ell^2+(\frac{d}{2})^2}$.</p>
<p>Therefore in your answer $d$ should be replaced by $2d$, giving $n=6$.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3372/find-minimum-distance-between-dipoles&show=3390#a3390Wed, 20 Feb 2019 21:03:31 +0000Find position and velocity at t=0?
http://physics.qandaexchange.com/?qa=3377/find-position-and-velocity-at-t-0
<p>Full question: </p>
<blockquote><p>An object moves with constant acceleration. At t= 2.50 s, the position of the object is x=2.00 m and its velocity is v= 4.50 m/s. At t= 7.00 s, v= -12.0 m/s. <br>
Find: <br>
(a) the position and the velocity at t= 0;<br>
(b) the average speed from 2.50s to 7.00 s, and <br>
(c) the average velocity from 2.50s to 7.00 s.</p>
</blockquote>
<p>I tried using the kinematic equations of motion for constant acceleration but my answers make no sense. </p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3377/find-position-and-velocity-at-t-0Tue, 05 Feb 2019 00:03:55 +0000Answered: Friction between two disks
http://physics.qandaexchange.com/?qa=616/friction-between-two-disks&show=3371#a3371
<p>acceleration of both the disc must be same since there is pure rooling <br>
and friction on smaller disc will be downward and on the bigger dics it will be upwards(Newton's 3rd law) and both will be equal in magnitude.<br>
therefore, writting the equation for bigger dics, f(2R)=2R²/2*(alpha)</p>
<pre><code> =>, f=2N
</code></pre>
<p>since the smaller dics has friction (f) as well as a external force due to which it is accelerating with 2m/s² and the external force was unknown initially. But now it also be found by writing the equation for smaller dics and putting f=2N.</p>
<pre><code> "sorry my previous answer was wrong"
</code></pre>
Physics Problemshttp://physics.qandaexchange.com/?qa=616/friction-between-two-disks&show=3371#a3371Fri, 01 Feb 2019 07:41:22 +0000Frequency modes of the rectangular shell
http://physics.qandaexchange.com/?qa=3357/frequency-modes-of-the-rectangular-shell
<p>This is task i received from my professor:</p>
<blockquote><p>The shapes of three natural modes having the frequencies $\omega_1, \omega_2, \omega_3$ of the rectangular shell are presented in the figure. The exciting pressure $p(t)$ applied uniformly all over the one side of the shell has the form $p(t) = Pe^{jωt}$. <br>
Make a sketch of the normal displacement of the gravity point of the shell against frequency, if the excitation frequency varies within bounds $0.5\omega_1< \omega <2\omega_1$ and static displacement of that point equals to $u_0$.</p>
</blockquote>
<p>Links to photo of frequency nodes (sorry for low quality)-><a rel="nofollow" href="https://ibb.co/b1nh3j9">1</a> .</p>
<p>Can somebody help me and tell me how i should get started?</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3357/frequency-modes-of-the-rectangular-shellMon, 14 Jan 2019 21:42:44 +0000Answered: tension in wires from which a block is suspended
http://physics.qandaexchange.com/?qa=3349/tension-in-wires-from-which-a-block-is-suspended&show=3354#a3354
<p>In order to compare $T$ and $W$, one needs to express one in terms of the other. As explained in the diagram itself, since the block is in translational equilibrium, we have:</p>
<p>$$\vec{T_1}+\vec{T_2}=\vec{W_b}$$</p>
<p>Projecting this relation onto the $Oy$ axis, the components of the tensions are both $T\sin\theta$ (because the angle between $T$ and $Oy$ is $\frac{\pi}{2}-\theta$ and the angle between $T$ and $Ox$ is $\theta$). Therefore:</p>
<p>$$2T\sin\theta=W\implies T=\dfrac{W}{2\sin\theta}$$</p>
<p>The information <em>the strings are almost horizontal</em> is equivalent to $\theta\cong 0$. For very small angles, the sine function is as well extremely small, and almost equal to the angle $\theta$ itself. Take a look at the following examples (I've included them here solely to make sure that you understand why $\sin\theta\to 0$ as $\theta\to 0$)</p>
<ul>
<li>$\sin(0.01)\cong 0.00999983333333\cong 0.01$</li>
<li>$\sin(0.0001) \cong 0.00009999999983\cong 0.0001$</li>
<li>$\sin(0)=0$</li>
</ul>
<p>So the denominator $2\sin\theta\cong0$ and therefore $T\to\infty$. As $W=mg$, clearly $W$ is a finite value (making the reasonable assumption that the mass is not inifite), and therefore <em>clearly</em> $T>W$. </p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3349/tension-in-wires-from-which-a-block-is-suspended&show=3354#a3354Fri, 11 Jan 2019 19:52:11 +0000How to solve this throwing ball off cliff question
http://physics.qandaexchange.com/?qa=3350/how-to-solve-this-throwing-ball-off-cliff-question
<p><img src="https://cdn.pbrd.co/images/HVdO2s0.png" alt=""></p>
<p>For b) I was curious why i couldn't use $s = ut + \frac12at^2$ as my equation as i had all the necessary known values. Instead to get the correct answer i had to use $v=u + at$.</p>
<p>Why is this so? Thank you for your time.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3350/how-to-solve-this-throwing-ball-off-cliff-questionSun, 06 Jan 2019 15:45:23 +0000Time period of small oscillation- disk and particle system
http://physics.qandaexchange.com/?qa=3346/time-period-of-small-oscillation-disk-and-particle-system
<p><img src="https://cdn.pbrd.co/images/HUWMQA2.jpg" alt=""></p>
<p>Attempt: </p>
<p>$y_{COM} = \dfrac{R}{3} = \dfrac {25}3$</p>
<p>$I = \dfrac 12 (2)R^2 + 1R^2 = 2\times 25 = 50 $</p>
<p>Now, T is given by: $T =2\pi \sqrt{\dfrac{I}{mgl}}$ where l is the distance of the centre of mass from the centre of rotation</p>
<p>So $T =2\pi \sqrt{\dfrac{50}{3 \times 10 \times \dfrac{25}{3}}} = 2\pi\sqrt{\dfrac{1}{5}}$</p>
<p>No option :( </p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3346/time-period-of-small-oscillation-disk-and-particle-systemFri, 04 Jan 2019 22:33:35 +0000Wave optics- number of minimas.
http://physics.qandaexchange.com/?qa=3345/wave-optics-number-of-minimas
<p><img src="https://cdn.pbrd.co/images/HUWujTQ.jpg" alt=""></p>
<p>Attempt: </p>
<p>Let m be the integer associated with $420 nm$<br>
and n be the integer associated with $540 nm$</p>
<p>$d \sin \theta = k \lambda $ $k\in Z$<br>
Clearly, </p>
<p>$m_{max} = 180$ </p>
<p>$n_{max} = 140 $</p>
<p>$(2m+1) \lambda_1 = (2n+1)\lambda_2$ (condition for dark fringes to overlap)</p>
<p>$\implies \dfrac{2m+1}{2n+1} = \dfrac 97$</p>
<p>$\implies m = \dfrac{1+7n +2n }{7}$</p>
<p>Hence, we obtain, for m to be an integer: $2n+1 = 7k$</p>
<p>$\implies n = \dfrac{7k-1}{2}$ where $k \in Z$</p>
<p>Now note that k must be odd since odd-1 = even </p>
<p>Thus, using $n \le 140$, $k_{max} = 39$</p>
<p>Now, we have to consider only odd values of k which are $1,3,...39$ = 20 numbers </p>
<p>Thus, we have 20 minimas on the upper side and 20 on the lower,<br>
Total minimas = $20+20 = 40$</p>
<p>But answer is $D$</p>
<p><strong>Question 1:</strong> What is wrong with my method? </p>
<p><strong>Question 2:</strong> Considering that this is a JEE Mains problem, what is the fastest 2 minute way to do it? </p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3345/wave-optics-number-of-minimasFri, 04 Jan 2019 22:01:11 +0000Answered: oblique collision of a rigid rod with a plane surface
http://physics.qandaexchange.com/?qa=3336/oblique-collision-of-a-rigid-rod-with-a-plane-surface&show=3344#a3344
<p>The normal contact impulse $J$ between the rod and plane has two effects : it changes the velocity of the CM of the rod, and it makes the rod rotate about its CM. </p>
<p>The impulsive force is related to the change in momentum of the CM of the rod, and the impulsive torque to the change in angular momentum of the rod, as follows : $$J=m(v_0+v_1)$$ $$J\frac{L}{2}\cos\theta=I\omega$$ Here $v_0, v_1$ are the vertical velocities of the CM before and after the collision, and $I=\frac{1}{12}mL^2$ is the moment of inertia of the rod about its CM. </p>
<p>The rod is not rotating initially. After the collision, the velocity of the end A which collided with the ground is $v_2'=\frac{L}{2}\omega$ relative to the CM of the rod, directed perpendicular to the rod, ie at an angle of $\theta=60^{\circ}$ to the vertical. The CM of the rod is moving upwards with speed $v_1$ so the vertical component of the velocity of A relative to the ground after collision is $$v_2=v_2'\cos\theta+v_1$$ </p>
<p>The elasticity of the collision is known ($e=1$) so the <strong>Law of Restitution</strong> can be applied to the relative velocities of approach and separation at the point of contact : $$v_2=ev_0$$</p>
<p>From the above equations you can eliminate $v_2, v_2', \omega$ to find the rebound speed $v_1$ of the CM of the rod and thereby the height to which the CM rises after collision.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3336/oblique-collision-of-a-rigid-rod-with-a-plane-surface&show=3344#a3344Wed, 02 Jan 2019 22:07:06 +0000Answered: Diffusion with reflecting boundary
http://physics.qandaexchange.com/?qa=3254/diffusion-with-reflecting-boundary&show=3343#a3343
<p>You have not gone wrong. You simply have not gone far enough.</p>
<p>You have obtained a quartic equation $$z^4-2z+1=0$$ This has 4 solutions, not only the obvious one $z=1$. The solution $z=1$ requires $t=+\infty$ so this is not the solution you require. You need to find some more solutions. Then check which one describes the situation you are trying to find.</p>
<p>In fact there is one other real solution and 2 imaginary solutions. Probably you should solve numerically (this is a physics question, not maths). </p>
<p>One simple method is to make a first guess $z<em>0$ then reiterate $$z</em>{n+1}=\frac12 (z_n^4+1)$$ With an initial guess of $z_0=0.5$ I get the following iterations :<br>
0.5<br>
0.53125<br>
0.5398259163<br>
0.5424604827<br>
0.543295467<br>
0.543562654<br>
0.5436484118<br>
0.543675964<br>
0.5436848186<br>
0.5436876646</p>
<p>These converge to 0.5437, though not quickly. </p>
<p>Alternatively apply the <a rel="nofollow" href="http://www.sosmath.com/calculus/diff/der07/der07.html">Newton-Raphson Method</a>. Using $$f(z)=z^4-2z+1=0$$$$f'(z)=4z^3-2$$ $$z_{n+1}=z_n-\frac{f(z_n)}{f'(z_n)}$$ again with $z_0=0.5$ we get<br>
0.5<br>
0.5416666667<br>
0.5436837222<br>
0.5436890127<br>
0.5436890127</p>
<p>which converges much more rapidly.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3254/diffusion-with-reflecting-boundary&show=3343#a3343Wed, 02 Jan 2019 16:42:47 +0000Answered: floating cubical block
http://physics.qandaexchange.com/?qa=3319/floating-cubical-block&show=3342#a3342
<p>I think you have misinterpreted the question. </p>
<p>The copper block is not hollow, it is solid copper. There is nothing inside it except copper. Mercury and water are not poured into the copper block. They are poured into a vessel in which the copper block is able to float.</p>
<p>Initially the copper block is floating inside a vessel (eg a beaker) which contains a layer of liquid mercury at the bottom. Some water is added on top of the mercury, surrounding the copper block, until the water is level with the top of the copper block. Now the copper block is floating in a layer of mercury and a layer of water. The question is asking for the depth of the layer of water. </p>
<p>The words "the copper in the block just gets submerged" in your question should read "the copper block just gets submerged".</p>
<hr>
<p>Mercury is a liquid. Water is a liquid. The LHS of your equation gives the pressure in the mercury level with the base of the copper block (neglecting air pressure). The RHS is the pressure due to copper block, which you can think of as another liquid.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3319/floating-cubical-block&show=3342#a3342Wed, 02 Jan 2019 15:26:29 +0000Answered: Cylinder lying on conveyor belt
http://physics.qandaexchange.com/?qa=3323/cylinder-lying-on-conveyor-belt&show=3340#a3340
<p>In essence this question has been answered already in <a rel="nofollow" href="http://physics.qandaexchange.com/?qa=470/rotation-in-truck">rotation of wheels and axle in accelerating truck</a>. There I show that the linear acceleration $a_0$ (relative to the ground) and the rotational acceleration $\alpha$ of a solid disk or cylinder which rolls without slipping on a lorry (or conveyor belt) which has translational acceleration $a$ are given by $$a_0=\frac13 a$$$$R\alpha=2a_0=\frac23 a$$ where $R$ is the radius of the cylinder. </p>
<p>The friction force is $$f=\frac13 Ma \le \mu Mg$$ so to avoid slipping the acceleration of the belt must be limited to $a \le 3\mu g$. </p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3323/cylinder-lying-on-conveyor-belt&show=3340#a3340Wed, 02 Jan 2019 08:31:03 +0000Answered: Final charge on the capacitor and Charge on capacitor as a function time
http://physics.qandaexchange.com/?qa=3311/final-charge-the-capacitor-charge-capacitor-function-time&show=3318#a3318
<p>The diagram is faulty. Switch S1 needs to be in the same branch as the capacitor to prevent it from discharging before S1 is closed. But we can guess what they mean.</p>
<p>Using KVL is not much more difficult than what you have done already. It should give you the correct answer - for example, as follows.</p>
<p>Mark 3 loop currents $i_1, i_2, i_3$ as shown in the diagram below. Then applying KVL around each loop we get $$\frac{Q}{C}=3Ri_1-Ri_2$$ $$-\frac{Q}{C}=4Ri_2-Ri_1-2Ri_3$$ $$E=4Ri_3-2Ri_2$$ We are interested in the current through the capacitor which is $i=i_1-i_2$. Substituting and eliminating $i_1, i_2, i_3$ we obtain <br>
$$i=-\frac{dQ}{dt}=\frac{1}{2CR}(Q-\frac14 CE)$$<br>
$$Q-\frac14 CE=(Q_0-\frac14 CE)e^{-t/2RC}$$ </p>
<p>After a long time ($t \to \infty$) the factor $e^{-t/2RC} \to 0$. Then the charge on the capacitor becomes $Q_{\infty}=\frac14CE$.</p>
<p><img src="https://i.imgur.com/y964baO.png" alt=""></p>
<p>A quicker solution is to find the equivalent resistance $R'$ across the capacitor after shorting the cell. The time constant of the decay (or increase) of charge on the capacitor is then $C R'$. If the initial charge is $Q_0$ and the final charge is $Q_1$ then the time-variation of the charge is given by </p>
<p>$$ Q(t) - Q_1 = (Q_0 - Q_1) e^{-t / C R' }$$ </p>
<p>After shorting the cell the circuit can be drawn as on the right above, because B and C are at the same potential. The resistance between C and D is $R$ because this is the result of two resistors $2R$ in parallel. Then the resistance along ADC is $2R$. This is the same as the resistance along ABC. ADC and ABC are in parallel, and have an equivalent resistance $R$. This is in series with the resistor in branch AC. So the effective resistance in series with the capacitor is $R'=2R$. The time constant for the circuit is $CR'=2CR$.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3311/final-charge-the-capacitor-charge-capacitor-function-time&show=3318#a3318Tue, 18 Dec 2018 03:46:46 +0000Answered: A projectile is ejected with vertical speed $v$ from the surface of a planet of mass $M$ and radius $R$
http://physics.qandaexchange.com/?qa=3295/projectile-ejected-with-vertical-speed-surface-planet-radius&show=3316#a3316
<p>Your mistake is that you assumed that the acceleration due to gravity $g$ is constant. This is only approximately true over distances which are very much smaller than the average distance of the object from the centre of the planet. The equation $y=vt-\frac12 gt^2$ only applies for constant acceleration $g$.</p>
<p>You need to use the conservation of energy. The initial KE plus gravitational PE at the surface of the planet is equal to the final KE plus gravitational PE at any other point such at that where it comes to rest instantaneously.</p>
<p>If the projectile were not fired vertically but instead at an angle to the vertical it would be necessary also to apply the conservation of angular momentum. In this problem the angular momentum is always zero.</p>
<p>If $v=c$ then Special Relativity Theory ought to apply instead of Newtonian Mechanics, so the kinetic energy would not be $\frac12 mv^2$. However this question assumes that Newtonian Mechanics is still valid when $v=c$. The tags for Special and General Relativity are not needed for this question.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3295/projectile-ejected-with-vertical-speed-surface-planet-radius&show=3316#a3316Sun, 16 Dec 2018 00:00:02 +0000Answered: Prove that the vector of acceleration always points towards the origin
http://physics.qandaexchange.com/?qa=3312/prove-that-vector-acceleration-always-points-towards-origin&show=3313#a3313
<p>This is a very mixed-up question.</p>
<p>Planets move in elliptical orbits. which can be described by the <strong>parametric equations</strong> $$x(t)=a\cos(kt), y(t)=b\sin(kt)$$ However, in these equations <strong>in general t is NOT time</strong>. It is just a parameter with no physical meaning. These equations correctly describe the path ($x,y$) taken by the planet, but not the relations ($x, t$) and ($y, t$) between its position and time. That relation is much more complex, and cannot be described by any simple function.</p>
<p>The only case in which parameter $t$ represents time for a gravitational orbit, for which the force has the form $-K/r^2$, is when the orbit is a circle, ie when $a=b$. Then $k$ is the angular velocity. In general (ie when $a\ne b$) the angular velocity is not $k$.</p>
<p>The equations do give the correct relation between position and time for an orbit of a mass on a spring with zero natural length, ie <strong>a Hooke's Law central force</strong> which has the form $-Kr$. However, even for such an orbit $k$ is not the angular velocity, which is not constant - unless the orbit is circular. </p>
<p><strong>How to show that the vector acceleration always points towards the origin</strong></p>
<p>The functions you have obtained for acceleration are the components of a vector : $$a_x=-k^2x, a_y=-k^2y$$ The magnitude of this vector is $$a=\sqrt{a_x^2+a_y^2}=-k^2r$$ where $r$ is the distance of the object from the centre of the ellipse. This confirms that the central force is proportional to $r$ - ie it is a Hooke's Law force. You need to find the direction of this vector. The gradient of the vector wrt the x axis is $$m=\frac{a_y(t)}{a_x(t)}=\frac{y}{x}$$ This vector passes through the point $(x, y)$ and its gradient is $y/x$. From this information you can conclude that <strong>the force is always directed towards the origin</strong> which is at the centre of the ellipse.</p>
<p>Although the orbit is an ellipse, <strong>it has no relation to a gravitational orbit</strong>, for which the force is always directed towards one focus of the ellipse, where the Sun is located, not towards the centre of the ellipse. </p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3312/prove-that-vector-acceleration-always-points-towards-origin&show=3313#a3313Sat, 15 Dec 2018 22:49:42 +0000Answered: System of two ideal Fermi gases in three dimensions.
http://physics.qandaexchange.com/?qa=3294/system-of-two-ideal-fermi-gases-in-three-dimensions&show=3310#a3310
<p>It is all about knowing that the ground-state pressure of the ideal Fermi gas is:</p>
<p>$$P_o = \frac{2}{5} n \epsilon_f$$</p>
<p>And actually (after some dimensional analysis), I realised that <strong>equation 19 is wrong</strong> as it should be:</p>
<p>$$p_o = \frac{2}{5}( \frac{3 h^3}{(2s+1)8 \sqrt{2} \pi m^{3/2}})^{2/3} n^{5/3}$$</p>
<p>From that point on, it drags that mistake.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3294/system-of-two-ideal-fermi-gases-in-three-dimensions&show=3310#a3310Thu, 13 Dec 2018 12:13:41 +0000Answered: Radial distribution of particle separation in a liquid at small distances
http://physics.qandaexchange.com/?qa=3233/radial-distribution-particle-separation-liquid-distances&show=3308#a3308
<p><strong>1. Preference</strong></p>
<p>The spatial distribution of particles settles into a configuration which minimises potential energy. This is achieved when there is some regular structure - for example, <strong>hexagonal close packing</strong>. Gravitational potential energy is far too small on the scale of microscopic particles. The balance is between kinetic energy and electrostatic potential energy as defined by the Lennard-Jones potential.</p>
<p>Ideal gas particles are assumed to have no attraction at all, only "hard sphere" repulsion when they come into contact. The potential function for ideal gas interactions is therefore $U(r)=0$ for $r>2r_0$ and $U(r)=∞$ for $r≤2r_0$, where $r_0$ is the radius of particles so $2r_0$ (twice particle radius) is the minimum distance between the centres of particles.</p>
<p><strong>2. The Nature of the Oscillations</strong></p>
<p>These are not <strong>dynamic</strong> oscillations in the motion of particles (periodic variations with time), like a mass on a spring. So the question about over/under-damping is not relevant - although it is possible some analogy could be made with dynamic oscillations. </p>
<p>These are <strong>static</strong> oscillations in the distribution of inter-particle distances (oscillations in space), like the ripples of a wave function in quantum mechanics, or the ripples in sand on a beach. </p>
<p>$g(r)$ is a probability distribution function. The "oscillations" indicate that there are periodic values of particle separation which are more likely than average (peaks) or less likely than average (troughs). These "oscillations" are not necessarily sinusoidal (harmonic).</p>
<p>High density causes the "oscillations" by restricting the space which particles can move about in. When particles are forced close together they slide into relative positions where they can keep as much motion as possible - ie they form structures such as hexagonal close packing. At low densities particles have plenty of freedom to move about and can occupy all relative positions equally. All values of separation $r$ are equally likely - the probability distribution function is uniform, flat.</p>
<p>The oscillations are not necessarily sinusoidal. The graph you linked does <strong>look</strong> like a damped harmonic oscillation. But it is not an oscillation in time and space like a pendulum, it is a periodic variation in the probability of finding a particle at distance $r$ from an arbitrary particle. It is probably not easy to see it if you look at the particles. It is a statistical variation which shows up when you calculate average the distances, because averaging removes random variations.</p>
<p>The explanation you quoted suggests that the graph is statistical. It has been constructed by taking a snapshot of the jiggling particles, and measuring the distance of every particle from one aribitrary particle, which is chosen as the origin. (Perhaps this is repeated with every particle in turn being used as the origin. This is the same as measuring the distance between every pair of particles.) The distances of every particle from the origin are measured, and a histogram is plotted of the <strong>density</strong> of particles at a distance in the range $r−Δr$ to $r+Δr$ (vertical axis) vs r (horizontal axis).</p>
<p>For a gas you would expect a uniform density distribution. </p>
<p>For a 'cold' 3D cubic crystalline solid you find sharp peaks at separations r which are multiples of $1,\sqrt2, \sqrt3, 2, \sqrt5, \sqrt6, ...$ units. In fact for all $r^2=l^2+m^2+n^2$ with all possible combinations of integers $l, m, n$. In between are ranges in $r$ which have zero frequency - eg 1.25 units. These peaks and troughs extend over a large range in $r$ - ie there is <strong>long range order</strong>. </p>
<p>For a 'hot' crystal these peaks and the troughs in between them are broader and more rounded, like rolling hills and valleys, giving the impression of oscillations. (The unit of spacing also increases as the solid expands.) </p>
<p>For a liquid or amorphous solid there are peaks (and troughs) only at small values of r - ie there is <strong>short range order</strong>. At large values of r the distribution is more uniform like a gas.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3233/radial-distribution-particle-separation-liquid-distances&show=3308#a3308Tue, 11 Dec 2018 20:14:41 +0000Check whether the followings are null, spacelike or timelike
http://physics.qandaexchange.com/?qa=3275/check-whether-the-followings-are-null-spacelike-or-timelike
<p>(a) <br>
$x^0=\int r^2+z^2d\tau$<br>
$x^1=\int r\sin\theta d\tau$<br>
$x^2=\int r\cos\theta d\tau$ <br>
$x^3=\int z d \tau$<br>
(b)<br>
$x^0= \sqrt3 ct$<br>
$x^1=ct$<br>
$x^2=ct_0 \sin(t/t_0)$<br>
$x^3=ct_0 \cos(t/t_0)$<br>
where $t_0$ is constant.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3275/check-whether-the-followings-are-null-spacelike-or-timelikeSun, 09 Dec 2018 11:01:29 +0000Answered: Rotation of L shaped rods after impulse
http://physics.qandaexchange.com/?qa=3304/rotation-of-l-shaped-rods-after-impulse&show=3305#a3305
<p>The hardest part of these questions is working out what they mean! Here it is not clear (i) whether joint B is free to translate or is fixed to the ground, and (ii) whether the rods are joined rigidly at B (so that they remain at $90^{\circ}$ at all times), or pinned loosely so the angle between them can change freely.</p>
<p>If the rods are joined rigidly at B then there will be a <strong>torque</strong> exerted on AB by rod BC as well as an impulsive force, and <em>vice versa</em> for AB acting on BC. This is because for two 2D objects to be attached rigidly together, and maintain the same relative positions, they must be pinned at 2 distinct points. The diagram suggests there is only one pin at joint B. Also, Part 2 of the question does not mention any torque. So I think the rods must be pinned loosely at B by a single pin. If this pin is also fixed to the ground then the impulse on AB will have no effect on BC because the reaction at the pin will be transmitted to the Earth, which is so massive that it does not move. </p>
<p>So I think<strong> joint B must be pinned loosely with a single pin and is free to translate</strong>. This makes sense because rods AB and BC can then have different angular velocities (see options in Part 1).</p>
<p>Your attempt assumes the two rods are rigidly connected, which is a reasonable interpretation of the question, but I think for the above reasons it must be wrong.</p>
<p>You are correct about the position of the COM of the 2 rods. But you have made an error in applying the <a rel="nofollow" href="http://hyperphysics.phy-astr.gsu.edu/hbase/parax.html">Parallel Axis Theorem</a>. This always starts with an axis through the COM and gives you the MI about an axis parallel to that through the COM. You have started at end B of the rod, not at the COM, so your calculation is incorrect. You should get $\frac{5}{12}m\ell^2$. </p>
<hr>
<p><strong>Solution</strong></p>
<p>Rod AB is struck at its COM. If it were not pinned to rod BC it would move upwards on the page, without rotating. This tells us that the reaction impulse $K$ at the pin is purely downwards on the page when acting on AB, and purely upwards when acting on BC.</p>
<p>Impulse $K$ acts through the COM of rod BC so this rod does not rotate : $\omega_2=0$. It moves upwards with velocity $v_2$ such that $K=mv_2$.</p>
<p>The impulses on AB can be resolved into a linear impulse $J-K$ acting upwards at the COM and a clockwise couple $K\frac{\ell}{2}$ about the COM. See <a rel="nofollow" href="https://physics.stackexchange.com/q/312523">Is net torque is not zero about all points on the rod for a linearly accelerating rod?</a> The COM of rod AB moves upwards with velocity $v_1$ such that $J-K=mv_1$. It also rotates with angular velocity $\omega_1=\frac{v_1-v_2}{ \ell/2}$ such that $K\frac{\ell}{2}=I_1\omega_1$ where $I_1=\frac{1}{12}m\ell^2$ is the moment of inertia of AB about its COM.</p>
<p>We now have enough equations so that we can find the following unknowns : $$\omega_1=\frac{6J}{5m\ell}, \omega_2=0$$ $$v_1=\frac{4J}{5m}, v_2=\frac{J}{5m}$$ $$K=\frac15 J$$</p>
<p>In Part 1 options A, B, C are correct, and in Part 2 option A is correct.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3304/rotation-of-l-shaped-rods-after-impulse&show=3305#a3305Sat, 08 Dec 2018 12:57:11 +0000Answered: The lean of a motorcyclist
http://physics.qandaexchange.com/?qa=3273/the-lean-of-a-motorcyclist&show=3276#a3276
<p>The back/front view should look like this :</p>
<p><img src="https://i.imgur.com/YGLurfc.png[/img]" alt=""></p>
<p>The vertical forces $N$ and $W$ cancel out. The resultant force is a radial friction force $f$ acting towards the centre of the turning circle. This is the centripetal force.</p>
<p>Another friction force acts tangentially, increasing the tangential velocity. This force is coming out of the page of the back/front view Free Body Diagram.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3273/the-lean-of-a-motorcyclist&show=3276#a3276Sun, 25 Nov 2018 00:56:03 +0000