Electric Field in Gaussian Spheres: Equilibrium and Charge Distribution

[antingtom]

Homework Statement

Consider a metal sphere and spherical shells. The innermost one is solid with radius R_1. A spherical shell surrounds the sphere and has an inner radius R_2 and an outer radius R_3. The sphere and shell are again surrounded by a shell of inner radius R_4 and outer radius R_5. None of the objects initially have a net charge. A negative charge -Q_0 is placed on the inner sphere and a positive charge +Q_0 is placed on the outermost shell.

i) After the charges have reached equilibrium, what will the direction of the electric field btwn the inner sphere and the middle shell?
ii) What will be the charge on the inner surface of the middle shell?
iii) What will be the charge on the outer surface of the middle shell?
iv) What will be the charge on the inner surface of the outermost shell?
v) What will be the charge on the outer surface of the outermost shell?
vi) What will the electric field plot look like?

Homework Equations

E.ds=Q(enclosed)/ε

The Attempt at a Solution

i) Does the fact that it's a metal sphere mean it's a conductor? If so, won't #1 be 0?
ii) inner surface of middle shell: E=1/4πε * -Q_0/(R_2)^2 ? (not 0 because of negative charge from inner sphere?)
iii) outer surface of middle shell: E=1/4πε * +Q_0/(R_3)^2 (I'm not sure if it would "feel" the Q_0 charge since r<R)
iv) inner surface of outermost shell: E=0
v) outer: E=1/4πε * +Q_0/(R_5)^2

I am really trying to understand this so I'd appreciate some help.

 

[Defennder]

i) Does the fact that it's a metal sphere mean it's a conductor? If so, won't #1 be 0?

They're asking for the E-field in between conductors metal sphere and metal shell. So it's not the E-field in a conductor

ii) inner surface of middle shell: E=1/4πε * -Q_0/(R_2)^2 ? (not 0 because of negative charge from inner sphere?)

They want the charge, not the E-field. Keep in mind the E-field in a conductor. Then apply Gauss law such the Gaussian surface is concentric within the middle conductor. What should the charge on the inner surface of the middle conductor be?

iii) outer surface of middle shell: E=1/4πε * +Q_0/(R_3)^2 (I'm not sure if it would "feel" the Q_0 charge since r<R)

This follows from the previous part. Now that you have the charge on the inner surface of the middle conductor, what charge would have to be on its outer surface?

iv) inner surface of outermost shell: E=0

This is almost the same as ii). Just use an appropriate gaussian surface.

v) outer: E=1/4πε * +Q_0/(R_5)^2

Use symmetry to deduce the direction of the E flux lines.

 

[baba89]

I would like to first clarify that the sphere and spherical shells mentioned in the problem are indeed conductors, as they are made of metal. This means that the charges placed on them will distribute themselves evenly on the surface of the objects, due to the repulsion between like charges.

Now, to answer the questions:

i) Since the charges have reached equilibrium, the electric field between the inner sphere and the middle shell will be zero. This is because the charges on the inner and outer surfaces of the middle shell will cancel out each other's electric field, resulting in a net electric field of zero.

ii) The charge on the inner surface of the middle shell will be -Q_0. This is because the negative charge placed on the inner sphere will induce a negative charge on the inner surface of the middle shell, due to the repulsion between like charges.

iii) The charge on the outer surface of the middle shell will be +Q_0. This is because the positive charge placed on the outermost shell will induce a positive charge on the outer surface of the middle shell, due to the attraction between opposite charges.

iv) As mentioned earlier, the charge on the inner surface of the outermost shell will be zero. This is because the positive charge on the outer surface of the middle shell will cancel out the negative charge on the inner surface of the outermost shell, resulting in a net charge of zero.

v) The charge on the outer surface of the outermost shell will be +Q_0. This is because the positive charge placed on the outermost shell will remain on its outer surface since there are no other charges to induce a different charge distribution.

vi) The electric field plot will look like a series of concentric circles, with the strongest electric field at the surface of the inner sphere and gradually decreasing as we move towards the outermost shell. The electric field will be zero between the inner sphere and the middle shell, as well as between the middle shell and the outermost shell, due to the cancelation of electric fields from the charges on their surfaces.