How Do You Calculate the Electric Potential of a Charged Cylinder?

[FS98]

Homework Statement

For the cylinder of uniform charge density in Fig. 2.26:
(a) show that the expression there given for the field inside the cylinder follows from Gauss’s law;
(b) find the potential φ as a function of r, both inside and outside the cylinder, taking φ = 0 at r = 0.


2. Homework Equations

The Attempt at a Solution

I finished part a and got the correct answers. I’m a bit confused about b now. Particularly the bit at the end about taking the potential and radius at 0. Can anybody explain where I start here?

 

[kuruman]

Use Gauss Law to find the electric field in the two regions (or read them off the figure) and then just integrate from the axis of the cylinder out.

 

[haruspex]

taking the potential and radius at 0.

Potential is always relative. There is, in principle, no absolute 0. In most electrostatics questions the convention is to set the potential to 0 at infinity, but in this case they are telling you to define the potential as zero at r=0. So the potential at infinity will not be zero.

 

[FS98]

then just integrate from the axis of the cylinder out.

Can you explain how and why this is done?

 

[kuruman]

Use the equation
$$V(r=B)-V(r=A)=-\int_A^B{\vec E \cdot d\vec r}$$
If you choose the potential to be zero at point A while B has some placeholder value r, then
$$V(r)-0=-\int_A^r{\vec E \cdot d\vec r}$$
Usually, the reference point A is taken at infinity. In this case, you are asked to take it at r = 0.