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Brazilian Center for Research in Physics

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Problem 57

(a) An insulating sphere with radius a has a uniform charge density $\rho$. The sphere is not centered at the origin but at $\vec{r} = \vec{b}$. Show that the electric field inside the sphere is given by $\overrightarrow{$} = \rho(\vec{r} - \vec{b} )/3\epsilon_0$ . (b) An insulating sphere of radius R has a spherical hole of radius a located within its volume and centered a distance b from the center of the sphere, where a $< b < R$ (a cross section of the sphere is shown in $\textbf{Fig. P22.57}$). The solid part of the sphere has a uniform volume charge density $\rho$. Find the magnitude and direction of the electric field $\overrightarrow{E}$ inside the hole, and show that $\overrightarrow{E}$ is uniform over the entire hole. [$Hint:$ Use the principle of superposition and the result of part (a).]

Answer

a) $E_{x}(0)=0$

b) $\overrightarrow{\boldsymbol{E}}=\left(\rho_{0} d / 3 \boldsymbol{\epsilon}_{0}\right)(x /|x|) \hat{i}$

c) $\overline{\boldsymbol{E}}=\left(\rho_{0} x^{3} / 3 \epsilon_{0} d^{2}\right) \hat{\boldsymbol{i}}$

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## Discussion

## Video Transcript

in this question. We have a solid insulating sphere off radios a and Villa Metric charge Density Rho, whose center is not located at the origin off the reference frame and have to calculate what is the electric field inside the sphere. For that, we begin by defining a new reference frame whose origin coincides with the center off that solves fear. So let me do it as follows we can define and rather y axis that I'm calling. Why prime such that it crosses the X axis precisely at the center off this fear like this. So this is my Y prime access. You can see that the origin off the reference frame defined by white, prime and ex coincides with the center off that salt sphere. Now we can easily calculate using gases Law Wadi ist the electric field inside that sphere. Gazans long tells us that the flux off electric field is given by the integral off D eight scaler Eat where D is a surface area element and E is the electric field. I noticed that there's a symmetry in this question because we have a sphere and this fear we produce electric field that points radio. The So It points like this always in radial directions. At the same time, you can see that the surface off that sphere has a normal vector. That is out, sir, at these directions. So D is a radio factor, and Suri's the electric field. It means that they are parole, so the eight is parallel to eat. And these implies that the eighth Scaler in is the Times E because they are part of vectors. Then you can remember that the magnitude off the electric field do not depend on the area off that Gulshan surface. Then it is egos through the magnitude of terrific field times, the integral off D over the area off the ocean surface. But the integral off D A. Over the area is just the surface area. Therefore, the foots off electric field is given by the magnitude off the electric field times the area off big ocean surface. Then the magnitude of the electric field is given by the flux, divided by the area off big ocean surface. But remember that the flux off the electric field through the surface off the Gulshan surface is also equals two. Then close a charge Q divided by absolute zero. So the magnitude off the electric field is given by the enclosed charge divided by the area of the ocean times. Absolute zero. But them what is then cause a charge? For that we have to peek a point. Suppose that we are working, for instance, In this point, we want to know what is the magnitude of the electric field. At this point, let me say that this point that a distance r from the origin so disease are Then we can pick big ocean surface that is, a sphere off radials are like this on. Then we can use the equation for the area office. Fear the area office Fear is given by four pi r squared then the electric field here has a magnitude given by Q. Divided by 45 R squared times. Absolute zero. Now we have to calculate what is cute. Q. Is the quantity of charge that is located inside off this guy ocean surface. Notice that we have a fella metric charge density Rho, which is uniforms the reform. We can use the fact that the charge is given by the charge density times. The volume in this case, we have to use the volume off the ocean surface, which is the volume office. Fear the value office Fear is given by four pi r two the third divided by treat Then then close it charge is rule times four pi aren't in the toilet divided by three, then a magnitude off the electric field Inside That fear is given by rope Times four by are to the third divided by three times four times Spy times are to the second absolute zero. Now we simplify whatever you can't So for planning for pie And here we have our tittered so we can simply fight We farmed in the second. Then we're last twist An electric field with a magnitude given by rope Times are divided by three times absolute zero This is the magnitude off the electric field in our reference frame. Why prime X? Because of the symmetry of this problem, it is easy to write this You know that perform. We know that the electric field will be parallel to the surface area of actor. So the electric field vector is given by three absolute zero times are so it is proportional to the event for our which points in the radial direction. Now, how can you translate these, Rector two of actor off. Why X Have to notice the following The only difference between why prime X and y axis is this vector. Let me call it the factor. Be so in order to bring these vectors to find it in white Prime X back to the origin off. Why X All we have to do Sue attract be from them. For instance, Here we have our So in order to bring this vector are back to the y X reference frame. We have to subtract b from it. So we end up with an electric field that is given by road divided by three times at zero times are minus beat the miners be term is what brings the vector back toe So original reference frame Now that Norland as the board so we can proceed to the next item in the next item we have these atop we have a big sphere off Radios are on uniformed volumetric charged into the room but it has a hole inside and it's whole is also a sphere off radios eight. That is not located at center off the bigs. From here, it's added the sense be from the center off that sphere. Then we have to complete what is the electric field in these region in the region off the whole. To solve this question, we can use the superposition principle off electrodynamics to treat the problem as follows. So instead of treating as fear withhold, we treat two spheres. One solid sphere off radios are in uniform. Votomatic charge density route that has no hole. Plus, as most fear, that is equivalent to the whole. So it's not located at the origin, but at a distance be from the origin. It has a radios A and very important. It has a volumetric charge. Density there is uniform and given by miners route so that when we super pose, this fear will be this year. What we got is the situation where we have a region inside the big sphere that has a net charge equals 20 and then the situation is very simple to treat. We already know what is the electric field. In the first situation. It's given by E one, and e one is echoes to roll, divided by three times absolute zero times are. Why do I know it? Why is it like this? I'm using this equation, but in this case the center off the sphere coincide with the origin off. The reference frame survey is close to zero and that's it. Nothing more. So this is the electric field. In the first situation for the second situation, the electric field is each you and it's given by miners rope divided by three times absolute zero times are miners meet. And then, using the superposition principle, we concluded that the electric field inside the whole E is equals two e one plus each year on Deasy's room, divided by three times absolute zero times are plus minus rope. Times are mine is me divided by three times absolute zero. These gives us rope. Times are divided by three times absolute zero miners. Roll times are divided by three times absolute zero. Plus I'm using the distributive property of multiplication Here roll times meat divided by three times absolute zero. We can see that there is a cancellation happening. So this term councils this turn and we end up with an electric field that is even by rope divided by three times absolute zero Pancks be so this is the electric field inside the whole notice that it only depends on B and B's a constant, so the electric field inside the hole is constant.

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