A nonconducting solid sphere has a uniform volume charge density ρLet r⃗ be the vector from the

A nonconducting solid sphere has a uniform volume charge...

A nonconducting solid sphere has a uniform volume charge density $\rho$ Let $\vec{r}$ be the vector from the center of the sphere to a gen- eral point $P$ within the sphere. (a) Show that the electric field at $P$ is given by $\overline{E}=\rho \vec{r} / 3 \varepsilon_{0}$ (Note that the result is independent of the ra- dius of the sphere.) (b) A spherical cavity is hollowed out of the sphere, as shown in Fig. $23-$ $60 .$ Using superposition concepts, show that the electric field at all points within the cavity is uniform and equal to $E=\rho \vec{a} / 3 \varepsilon_{0},$ where $\vec{a}$ is the position vector from the center of the sphere to the center of the cavity.

A nonconducting solid sphere has a uniform volume charge density $\rho$ Let $\vec{r}$ be the
vector from the center of the sphere to a gen-
eral point $P$ within the sphere. (a) Show that
the electric field at $P$ is given by $\overline{E}=\rho \vec{r} / 3 \varepsilon_{0}$
(Note that the result is independent of the ra-
dius of the sphere.) (b) A spherical cavity is
hollowed out of the sphere, as shown in Fig. $23-$
$60 .$ Using superposition concepts, show that
the electric field at all points within the cavity
is uniform and equal to $E=\rho \vec{a} / 3 \varepsilon_{0},$ where $\vec{a}$ is the position vector
from the center of the sphere to the center of the cavity.

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the net electric flux through a closed surface to the charge enclosed by that surface. This concept is especially useful when dealing with problems involving symmetric charge distributions, as it allows for the straightforward calculation of electric fields within or outside of such distributions.

Uniform Charge Distribution

A uniform charge distribution refers to a scenario where the charge density remains constant throughout a given volume. This uniformity simplifies the analysis of the electric field, particularly in symmetric geometries such as spheres, as it leads to predictable and often linear variations in the field with respect to the distance from the center of the distribution.

Electric Field in a Charged Sphere

In a uniformly charged sphere, the electric field inside the sphere varies linearly with the distance from the center, as derived from Gauss's Law. This linear behavior arises because the enclosed charge is proportional to the volume of the sphere up to any point within it, resulting in a straightforward relationship between the electric field and the radial position.

Superposition Principle

The superposition principle states that the net electric field produced by a collection of charges is the vector sum of the individual fields produced by each charge. This concept is crucial when dealing with complex charge distributions or when modifications are made, such as the introduction of a cavity, as it allows for the analysis of the overall effect by considering the contribution of each part of the system independently.

Transcript

00:01 All right, so i remember seeing this one back in the day, and i was like, what? so you're given a sphere, you're told it has a cavity, and you want to get, ultimately you want to get the goal, the field inside that cavity.

00:23 And so first you want to just get the field inside here and show that it can be written in this nice way row over three epsilon not.

00:33 Times r.

00:34 So let's start with gauss's law, so not e equals, but the flux is equal to the q enclosed over epsilon not, and if we do some gaussian surface of radius r, then we can say e times pi r squared is equal to the charge enclosed over epsilon not.

00:57 The charge enclosed is going to be the total charge, so let q be the total, q equal total charge.

01:10 And only a portion of that total charge is going to be enclosed.

01:13 In particular, it's the ratio of the radius of this sphere with radius r to the radius of the full sphere, which i'll call big r.

01:23 And so that's going to be, so it'll be four thirds pi, little r cubed, over four thirds pi, big r cubed.

01:30 And if you can't solve the four thirds pi, you'll get this.

01:34 And then if you also use the definition of density, cube, so charges density times volume and then the volume and then the volume of the sphere is four thirds pi r cubed.

01:54 So if you sub in this q canceled this r, oops, i almost forgot my four here.

02:00 And then these fours will end up canceling.

02:02 You'll get the answer.

02:03 You'll get that e is equal to row over three epsilon knot.

02:10 And it's times r in the r -hat direction, which just gives you a vector r.

02:14 So now i'll do this on another page.

02:18 You make a cavity, and you're like, what's the field in the cavity? and you have to show that it's this constant, which is kind of interesting, that it comes out of a constant, and maybe that's something that i would ponder with you if i were actually here in real life with you, but i'm trying to make these videos not too long.

02:39 So, but do ponder it.

02:42 Maybe it'll be fun.

02:44 So anyway, so basically we know that the field and, we know the field and without the cavity is this row over three epsilon not times r.

02:54 We want to get it with the cavity.

02:55 So the trick to modeling the cavity is to say that it's the same thing as taking this full sphere and superimposing on it a sphere of negative charge.

03:06 Right, because the negative charges from this new sphere plus the, positive charges that were already there.

03:12 Maybe i'll kind of draw this in.

03:15 We'll make a cavity.

03:17 And then this is my representation of the charges that were already in the sphere.

03:22 And i didn't draw it this way, but they should be the same density of charges...

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