(a) An insulating sphere with radius a has a uniform charge density ρ. The sphere is not centere

(a) An insulating sphere with radius a has a uniform charge...

(a) An insulating sphere with radius $a$ has a uniform charge density $\rho .$ The sphere is not centered at the origin but at $\overrightarrow{\boldsymbol{r}}=\overrightarrow{\boldsymbol{b}}$. Show that the electric field inside the sphere is given by $\overrightarrow{\boldsymbol{E}}=\rho(\overrightarrow{\boldsymbol{r}}-\overrightarrow{\boldsymbol{b}}) / 3 \epsilon_{0} .$ (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 Fig. P22.53). The solid part of the sphere has a uniform volume charge density $\rho$. Find the magnitude and direction of the electric field $\overrightarrow{\boldsymbol{E}}$ inside the hole, and show that $\overrightarrow{\boldsymbol{E}}$ is uniform over the entire hole. [Hint: Use the principle of superposition and the result of part (a).]

(a) An insulating sphere with radius $a$ has a uniform charge density $\rho .$ The sphere is not centered at the origin but at $\overrightarrow{\boldsymbol{r}}=\overrightarrow{\boldsymbol{b}}$. Show that the electric field inside the sphere is given by $\overrightarrow{\boldsymbol{E}}=\rho(\overrightarrow{\boldsymbol{r}}-\overrightarrow{\boldsymbol{b}}) / 3 \epsilon_{0} .$ (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 Fig. P22.53). The solid part of the sphere has a uniform volume charge density $\rho$. Find the magnitude and direction of the electric field $\overrightarrow{\boldsymbol{E}}$ inside the hole, and show that $\overrightarrow{\boldsymbol{E}}$ is uniform over the entire hole. [Hint: Use the principle of superposition and the result of part (a).]

Key Concepts

Coordinate Translation (Displaced Charge Distributions)

In many electrostatic problems, the center of the charge distribution might not coincide with the origin of the coordinate system. The method of coordinate translation involves shifting the reference frame so that the effects due to displacement are properly accounted for. This is important when the charged object is not centered at the origin, as it affects the spatial dependence of the electric field.

Method of Superposed Charge Distributions for Cavities

The method of superposition can also be applied by considering a cavity within a charged body as being equivalent to superposing a region of negative charge density over the uniform positive charge distribution. This approach allows us to calculate the resulting uniform electric field in the cavity by subtracting the field due to the missing charge, demonstrating that under certain conditions the field inside the cavity is uniform.

Superposition Principle

The principle of superposition states that the net electric field produced by multiple charge distributions is the vector sum of the fields produced by each charge distribution alone. This concept is crucial when dealing with composite problems, such as a sphere with a spherical cavity, by conceptually ‘adding’ and ‘subtracting’ charge distributions to simplify the analysis.

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed within that surface. It is particularly useful for finding electric fields in problems with high symmetry, such as those involving spherical or cylindrical distributions of charge. The symmetry of a uniformly charged sphere allows us to use Gauss's Law to derive a simple expression for the electric field within the sphere.

Uniform Charge Distribution in a Sphere

A uniformly charged sphere is a standard model in electrostatics where the charge is evenly distributed throughout the volume of the sphere. The electric field inside such a sphere is known to vary linearly with the distance from the center. This concept is fundamental when analyzing fields due to volume charge distributions and forms the basis for more complex configurations involving displacements or cavities.

Transcript

0:00 In this question.

00:01 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.

00:17 For that, we begin by defining a new reference frame whose origin coincides with the center off that solves fear.

00:25 So let me do it as follows we can define and rather y axis that i'm calling.

00:31 Why prime such that it crosses the x axis precisely at the center off this fear like this.

00:41 So this is my y prime access.

00:45 You can see that the origin off the reference frame defined by white, prime and ex coincides with the center off that salt sphere.

00:54 Now we can easily calculate using gases law wadi ist the electric field inside that sphere.

01:01 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.

01:18 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.

01:26 The so it points like this always in radial directions.

01:31 At the same time, you can see that the surface off that sphere has a normal vector.

01:37 That is out, sir, at these directions.

01:40 So d is a radio factor, and suri's the electric field.

01:46 It means that they are parole, so the eight is parallel to eat.

01:51 And these implies that the eighth scaler in is the times e because they are part of vectors.

02:00 Then you can remember that the magnitude off the electric field do not depend on the area off that gulshan surface.

02:08 Then it is egos through the magnitude of terrific field times, the integral off d over the area off the ocean surface.

02:17 But the integral off d a.

02:19 Over the area is just the surface area.

02:22 Therefore, the foots off electric field is given by the magnitude off the electric field times the area off big ocean surface.

02:32 Then the magnitude of the electric field is given by the flux, divided by the area off big ocean surface.

02:42 But remember that the flux off the electric field through the surface off the gulshan surface is also equals two.

02:49 Then close a charge q divided by absolute zero.

02:53 So the magnitude off the electric field is given by the enclosed charge divided by the area of the ocean times.

03:01 Absolute zero.

03:03 But them what is then cause a charge? for that we have to peek a point.

03:09 Suppose that we are working, for instance, in this point, we want to know what is the magnitude of the electric field.

03:17 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.

03:34 Then we can use the equation for the area office.

03:38 Fear the area office fear is given by four pi r squared then the electric field here has a magnitude given by q.

03:48 Divided by 45 r squared times.

03:52 Absolute zero.

03:53 Now we have to calculate what is cute.

03:56 Q.

03:56 Is the quantity of charge that is located inside off this guy ocean surface.

04:02 Notice that we have a fella metric charge density rho, which is uniforms the reform.

04:08 We can use the fact that the charge is given by the charge density times.

04:14 The volume in this case, we have to use the volume off the ocean surface, which is the volume office.

04:20 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.

04:51 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...

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