Electric fields and electric potential

[Pioneer]

I've been doing some questions and I'm completely stuck on three of them. I tried thinking of how to tackle them but I'm coming up blank.

1. A charge of q is distributed uniformly throughout a spherical volume of radius R. Setting V = 0 at infinity, show the potential at a distance r from the center, where r<R, is given by
V = (q(3R^2 - r^2))/8piEoR^3).

2. A pendulum is hung from the higher of two large horizontal plates. The pendulum consists of a small conducting sphere of mass m and charge +q and an insulating thread of length L. What is the period of the pendulum of a uniform electric field E is set up between the plates by

(a) charging the top plate negatively and the lower plate positively
(b) and vice versa?

3. A uniform upward electric field E of magnitude 2.00 x 10^3 N/C has been set up between two horizontal plates by charging the lower plate positively and the upper plate negatively. The plates have length L = 10.0cm and separation d = 2.00cm. An electron is then shot between the plates from the left edge of the lower plate. The intial velocity of the electron makes an angle 45 degrees with the lower plate and has a magnitude 6.00 x 10^6 m/s.

(a) Will the electron strike one of the plates?
(b) If so, which plate and how far horizontally from the left edge will the electron stike?

 

[Lyuokdea]

what work have you done on these so far? Most people will want to see some work before they help you too much, here is a hint for number 1 though:

First, there should be some more specification in the question about the V=0 part, V is a function of x, is it V(0) = 0 or V(infinity) = 0 ? You can pick either one without any problems, because only the change in potential matters, however, you will get different forms of answers. It looks like in this problem you want to set V(0) = 0.

So now, look at the definition that you have for potential. And also use your knowledge of Gauss's Law in order to allow you to determine the charge that is contained when you are at any radial point on the sphere.

~Lyuokdea

 

[Pioneer]

So here's what I attempted with question 1

For r > R the sphere behaves as a point charge

V(r) = Q/4piEor therefore at the surfaces, potential V(R) = Q/4piEor

Va - Vb = Q/4piEor^3 [(rb^2/2) - ra^2/2)]

for rb= R, Vb = Q/4piEoR

Va = Q/4piEoR = Q/8piEor^3 (R^2 - ra^2) + 2Q/8piEoR

that is Va = Q/8piEoR - (Qra^2/8piEor^3) + 2Q/8piEoR