Electrostatic Potential for Two Concentric Cylindrical Shells

[sweetdion]

Homework Statement

Two very long hollow conducting cylindrical shells are situated along the x-axis. The shells are concentric and have negligible thickness. The inner shell has a radius a and a linear charge density +lambda, while the outer shell has a radius b and a linear charge density -lambda. Take the zero of electrostatic potential to be at r = 0. The coordinate r measures the distance from the common axis of the two cylinders in a region far from either end.

a) Determine the electrostatic potential V(r) for all values of r
b) Sketch V(r) vs. r for all r
c) Determine the potential difference DeltaV between r=a and r=b
d) If a positive charge +q is released from rest at r=a, what will be its kinetic energy when it reaches the outer cylinder at r=b.

Homework Equations

V(r)=q/4piE0r
Delta V=Vb-Va=q/4PiE0*(1/rb-1/ra)
Ke=1/2mv^2
U=kq1q2/r
KEf=Ui

The Attempt at a Solution

a) at r=a
V=q/4piE0a

at r=b
V=q/4piE0b

b) as the radius increases the potential goes down. It starts at some positive y value and ends at some negative y value.

c)Delta V=Vb-Va=q/4PiE0*(1/b-1/a)

d) U=kq1q2/a
1/2mv^2=kq1q2/r
v^2=2kq1q2/mr
v=(2kq1q2/mr)^1/2

 

[sweetdion]

I'm just wanting to know if i got this right

 

[ehild]

Homework Statement

Two very long hollow conducting cylindrical shells are situated along the x-axis. The shells are concentric and have negligible thickness. The inner shell has a radius a and a linear charge density +lambda, while the outer shell has a radius b and a linear charge density -lambda. Take the zero of electrostatic potential to be at r = 0. The coordinate r measures the distance from the common axis of the two cylinders in a region far from either end.

Homework Equations

V(r)=q/4piE0r

The formula you quoted refers to a point charge. These are very long cylinders.

ehild

 

[sweetdion]

V(r)= 1/4piE0 Int dq/r

 

[sweetdion]

Part C)

EA=Qenc/E0
E=lamda/PirE0
Delta V=-lamda/PiE0 Int dr/r
Delta V= lamda/PiE0 ln(b/a)

 

[ehild]

You have started part C well, but it is still wrong.

The first questions were:

"a) Determine the electrostatic potential V(r) for all values of r
b) Sketch V(r) vs. r for all r"

So what is the potential for r<a? for r>b?

ehild

 

[sweetdion]

what equation am i supposed to use to figure out the potential? so confused.

V (r) = 1/4PiE0 * Int dq/r ?

 

[ehild]

Find the electric field first. What is it inside the inner cylinder? Between a and b? for r>b? Use Gauss' law.
For r<0, the enclosed charge is 0. If a<r<b, the enclosed charge is +lambda *Length of the cylinder. For a cylinder with r>b, the enclosed charge is 0. E is the negative gradient of the potential. What is the potential like if E=0?

ehild