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 center. The total charge on the shell is $-3 Q$, and it is insulated from its surroundings (Fig. P22.42). (a) Derive expressions for the electric-field magnitude $E$ in terms of the distance $r$ from the center for the regions $r<a, a<r<b,$ and $r>b$. What is the surface charge density (b) on the inner surface of the conducting shell; (c) on the outer surface of the conducting shell? (d) Sketch the electric field lines and the location of all charges. (e) Graph $E$ as a function of $r$.

A conducting spherical shell with inner radius $a$ and outer radius $b$ has a positive point charge $Q$ located at its center. The total charge on the shell is $-3 Q$, and it is insulated from its surroundings (Fig. P22.42). (a) Derive expressions for the electric-field magnitude $E$ in terms of the distance $r$ from the center for the regions $r<a, a<r<b,$ and $r>b$. What is the surface charge density (b) on the inner surface of the conducting shell; (c) on the outer surface of the conducting shell?
(d) Sketch the electric field lines and the location of all charges.
(e) Graph $E$ as a function of $r$.

Key Concepts

Electric Field Lines and Graphical Analysis

Electric field lines provide a visual representation of the direction and strength of the electric field. They originate from positive charges and terminate on negative charges, with their density indicating the field's magnitude. Graphical analysis, including plotting the electric field as a function of distance, is important for interpreting the behavior of the field across different regions and at the boundaries between conductors and surrounding space.

Spherical Symmetry

Spherical symmetry implies that the physical properties of the system depend solely on the radial distance from the center. This symmetry allows the use of spherical Gaussian surfaces, which greatly simplifies the application of Gauss's Law in deriving the electric field. It also ensures that the electric field points radially, avoiding complications that arise from directional variations.

Surface Charge Density

Surface charge density is a measure of the amount of charge per unit area on a surface. In the context of conductors, knowing the induced charges on the inner and outer surfaces helps determine the discontinuities in the electric field at these boundaries. Calculating the surface charge density is essential for understanding how the total charge on a conductor is distributed in response to enclosed charges.

Electrostatic Equilibrium in Conductors

In electrostatic equilibrium, conductors exhibit a zero electric field within their bulk. Charges in a conductor reside on its surfaces, and any excess charge distributes itself in such a way as to cancel the interior field. This principle is key to understanding how induced charges appear on the inner and outer surfaces of a conducting shell when exposed to external or enclosed charges.

Gauss's Law

Gauss's Law is a fundamental tool in electrostatics that relates the electric flux through a closed surface to the charge enclosed by that surface. It simplifies the calculation of electric fields, especially in situations with high symmetry such as spherical geometries. By selecting an appropriate Gaussian surface, one can derive expressions for the electric field in different regions based solely on the net charge enclosed.

Transcript

00:01 Welcome to the problem with twenty two point four four.

00:04 This is for you to go slow.

00:07 So here we have a conducting spherical shell within the radius a and outer radius b.

00:14 And there's a point charge located metis born charges capital cube.

00:20 So let me practice one.

00:22 So this is really cute.

00:27 It is a capital q charlie at the center and the older or the outer spectacle shell.

00:33 This has minus three.

00:35 Q.

00:35 Just hold on if you look at it carefully.

00:39 So this is conducting shin, and this is prosecutor has definitely minus.

00:43 Q.

00:43 Charles will be on the inner surface, and the remaining one would be madness to kill the old room.

00:50 So this minus q and minus two q will make minus tricky.

00:56 So which is correct? according to the given conditions.

01:01 Now we need to find an electric field between all regions, so first finding or directed field for are less than a so looking at this one, we have making cautions office.

01:19 If you want to throw the ocean surface, so having ideas are there is only plus future, so definitely it would be one over four pi ups or not, can't look you over.

01:32 Ask now going to the second part.

01:38 So we have any gin between our yes, me and left, which is hear that.

01:47 So this is a conducting material.

01:50 So we know that inside any conducting material that we feel is always so in this case too.

01:57 So we have elected field jesus now going to the party.

02:03 The party, the party, eh, with arthur off park dae.

02:10 So we need to find out the electric field at our grid of the sea.

02:13 So looking at some point here, so we have the stadium's what? and we make a go shin's office again having spectacle ship.

02:33 If you want to go, you can trouble not to confuse it.

02:36 So that's keep it that way.

02:42 So the outer shell has minus two.

02:45 Futures in a shell is minus q.