A conducting spherical shell with inner radius a and outer...

A conducting spherical shell with inner radius $a$ and outer radius $b$ has a positive point charge $Q$ located at its centre. The total charge on the shell is $-3 Q$, and it is insulated from its surroundings (Fig. 22.37). (a) Derive expressions for the electric-field magnitude in terms Figure 22.37 Problem 22.44. of the distance $r$ from the centre for the regions $r<a, a<r<b$ and $r>b$. (b) What is the surface charge density on the inner surface of the conducting shell? (c) What is the surface charge density on the outer surface of the conducting shell? (d) Sketch the electric field lines and the location of all charges. (e) Graph the electric-field magnitude as a function of $r$. (FIGURE CANT COPY)

A conducting spherical shell with inner radius $a$ and outer radius $b$ has a positive point charge $Q$ located at its centre. The total charge on the shell is $-3 Q$, and it is insulated from its surroundings (Fig. 22.37). (a) Derive expressions for the electric-field magnitude in terms
Figure 22.37
Problem 22.44. of the distance $r$ from the centre for the regions $r<a, a<r<b$ and $r>b$. (b) What is the surface charge density on the inner surface of the conducting shell? (c) What is the surface charge density on the outer surface of the conducting shell? (d) Sketch the electric field lines and the location of all charges. (e) Graph the electric-field magnitude as a function of $r$.
(FIGURE CANT COPY)

Key Concepts

Surface Charge Density

Surface charge density is a measure of how much charge resides per unit area on a surface, often denoted by ?. It is determined by considering the discontinuity in the electric field across a charged surface as described by boundary conditions in electrostatics. This concept is crucial for understanding how charge distributes itself on the boundaries of conductors and other materials.

Visualization of Electric Field Lines

Electric field line diagrams provide a graphical representation of the electric field in space, illustrating both the direction and the relative strength of the field. They help in visualizing how the field emanates from positive charges and terminates on negative charges, and they serve as useful tools for understanding charge distributions, especially in systems with induced charges or complex geometries.

Electric Fields in Spherical Symmetry

Electric fields generated by spherically symmetric charge distributions can be analyzed by considering different regions based on distance from the center of symmetry. In such setups, the field may vary distinctly in the innermost region, within a shell, and outside the shell. The symmetry simplifies the calculations and leads to expressions that depend solely on the radial distance from the center.

Charge Induction in Conductors

Charge induction refers to the process whereby a conductor develops regions of positive and negative charge in response to an external electric field or nearby charge. The presence of a charge within or near a conductor causes the free charges in the conductor to rearrange, leading to induced charges on its surfaces. This phenomenon is essential when analyzing systems where inner and outer surface charges differ due to external and internal influences.

Gauss's Law

Gauss's Law is a fundamental principle in electrostatics that relates the electric flux through a closed surface to the charge enclosed by that surface. It is particularly powerful in situations with high symmetry, such as spherical, cylindrical, or planar systems, allowing one to derive expressions for the electric field by choosing an appropriate Gaussian surface and simplifying the integral through symmetry arguments.

Electrostatic Equilibrium in Conductors

In electrostatic conditions, conductors exhibit a crucial property where the electric field inside the conducting material is zero. This results from the movement of free charges within the conductor, which redistribute themselves to cancel any applied fields. Understanding this concept is key to determining where and how charges reside on the surfaces of conductors.

Transcript

00:01 In this example, we're going to be determining the electric field at all points in space for the following configuration.

00:08 Okay, what we have is a conducting spherical shell with a point charge q at its center.

00:15 So here's q at the center and then that's surrounded by a conducting spherical shell.

00:27 Okay, it's got inner radius a, outer radius b.

00:37 Okay, this is supposed to be a circle, but you can see that it's not quite a circle, but we're going to pretend it's perfectly spherical.

00:44 Okay, this shell has a charge on it as well and this charge is going to be equal to negative three q.

00:54 Okay, this charge in the center has charge q.

01:00 So i want to find the electric field in three regions.

01:03 I want to find it for r less than a, k, r in between a and b, so less than b but greater than a.

01:19 And then i want to find it for r is greater than a.

01:25 Okay, so first we're going to be looking at the region inside the hollow spherical shell, the outside of that charge q in the center.

01:36 So this will be my first region.

01:39 Then our second region is from a to b, so we're inside the shell.

01:43 And then my third region, this is region two, my third region is everything outside of the shell.

01:50 Okay, so let's start calculating these fields.

01:54 We're going to be using gauss's law for all three locations.

02:01 Okay, gauss's law says that the e dot da equals q enclosed over epsilon naught.

02:12 Okay, so for my first region when we're inside this hollow spherical shell, my q enclosed is just q.

02:24 My surface integral, that's just e is perpendicular to the surface everywhere because we have that spherical symmetry so we can use gauss's law.

02:34 So e dot da is just e times the surface of the sphere for pi r squared.

02:42 Okay, so e for r less than a equals q over four pi epsilon naught r squared.

02:59 Okay, now we want to look at region two.

03:02 That's between the two walls of the shell.

03:08 Okay, the shell is a conducting shell.

03:10 What do we know about conductors? we know that there's no electric field within them.

03:15 Okay, all of the charge resides on the surface.

03:18 Okay, the two surfaces.

03:20 So we're going to have, we have a positive charge in the center.

03:23 So we're going to have negative charge distributed on the inner surface of the shell.

03:27 We'll have positive distributed on the outside.

03:31 Okay, so for region two, e equals zero.