(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 $\vec{r}=\vec{b} .$ Show that the electric field inside the sphere is given by $\vec{E}=\rho(\vec{r}-\vec{b}) / 3 \epsilon_{0} \cdot(\mathrm{b})$ 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. $\mathrm{P} 22.61$ ). The solid part of the sphere has a uniform volume charge density $\rho .$ Find the magnitude and direction of the electric field $\vec{E}$ inside the hole, and show that $\vec{E}$ 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 $\vec{r}=\vec{b} .$ Show that the electric field inside the sphere is given by $\vec{E}=\rho(\vec{r}-\vec{b}) / 3 \epsilon_{0} \cdot(\mathrm{b})$ 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. $\mathrm{P} 22.61$ ).
The solid part of the sphere has a uniform volume charge density $\rho .$ Find the magnitude and direction of the electric field $\vec{E}$ inside the hole, and show that $\vec{E}$ is uniform over the entire hole. [Hint: Use the principle of superposition and the result of part (a).

Key Concepts

Gauss's Law

Gauss's Law is a fundamental law in electromagnetism that relates the electric flux through a closed surface to the charge enclosed by that surface. It is particularly useful for calculating electric fields when the charge distribution possesses a high degree of symmetry, such as spherical symmetry, by allowing one to select a Gaussian surface where the electric field is either constant or zero.

Uniform Charge Distribution

A uniform charge distribution implies that the charge density is constant throughout the volume of the object. This simplifies the computation of the electric field, because it means that the total charge enclosed within any chosen volume is directly proportional to that volume, which is particularly useful when applying Gauss's Law to symmetrical problems like that of a uniformly charged sphere.

Electric Field Inside a Uniformly Charged Sphere

In a uniformly charged insulating sphere, the electric field inside the sphere is derived by applying Gauss's Law with a spherical Gaussian surface. The field at an internal point varies linearly with the distance from the center of the sphere due to the volume proportionality of the enclosed charge. When the sphere's center is displaced, the electric field is expressed in terms of the displacement vector, modifying the radial dependence accordingly.

Superposition Principle

The superposition principle states that the total electric field created by multiple charge distributions is the vector sum of the fields due to each individual charge distribution. This principle is particularly useful in solving complex problems where the charge distribution can be seen as a composition of simpler parts, such as a full sphere and a removed spherical portion (hole), allowing for the independent calculation of their electric fields and subsequent combination.

Coordinate Transformation and Displacement Effect

When dealing with objects whose centers are displaced from the origin, the calculation of the electric field requires a transformation of coordinates. This means that the reference point for the electric field is shifted by the displacement vector, and the resulting field expressions account for the difference between the position vector of the point of interest and the shifted center of the charge distribution. This concept is crucial in understanding problems where the physical position of the charge distribution does not coincide with the coordinate system's origin.

Transcript

00:01 For this problem on the topic of kauser's law, we are told that an insulating sphere with radius a has uniform charge density.

00:08 The sphere is centered at r is equal to b, and we want to show the electric field inside the sphere is given by the expression provided.

00:15 We are then told that an insulating sphere of radius capital r has a spherical whole of radius little a located within its volume and centered at b from the center of the sphere.

00:27 Now, if this solid part of the sphere has a uniform volume charge density row, we want to find the magnitude and direction of the electric field inside the hole and show that this electric field is uniform over the entire hole.

00:40 So firstly, for part a, we know that for an insulating sphere of uniform charge density row and centered at the origin, the electric field inside the sphere e is given by the total charge q times r prime.

00:59 Over 4 pi epsilon 0 times capital r cubed where r prime here is the vector from the center of the sphere to the point where the electric field e is calculated now we also know that the uniform charge density row is equal to 3 times q over 4 pi r cubed so we can write the electric field strength e as simply row times r over 3 epsilon 0.

01:39 But we need to write this as a vector.

01:41 We know the electric field vector is readily outward in the direction of r prime.

01:47 So the electric field vector e is equal to row times vector r prime over 3 epsilon not.

01:58 Now for a sphere whose center is located by vector b, a point inside the sphere is located by and r is located by the vector r prime is equal to r minus b relative to the center of the sphere as shown in the following figure.

02:18 So we can clearly see that for a sphere centered at point b, the electric field e is equal to row into r minus b divided by three times epsilon not...

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