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

• FS98
In summary, the conversation discusses using Gauss's law to find the electric field inside a cylinder with uniform charge density and then using that to find the potential as a function of r, both inside and outside the cylinder, taking the potential to be zero at r = 0. This involves integrating from the axis of the cylinder out and using the equation V(r=B)-V(r=A)=-∫A^B(E⃗ ⋅dr⃗ ). This convention of setting the potential to zero at a certain point is usually done at infinity, but in this case, it is set at r = 0.
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?

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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.

FS98
FS98 said:
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
kuruman said:
then just integrate from the axis of the cylinder out.
Can you explain how and why this is done?

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.

FS98

1. What is the potential of a charged cylinder?

The potential of a charged cylinder is a measure of the electrical potential energy per unit charge at any point outside the cylinder. It is typically expressed in volts and is dependent on the charge and size of the cylinder, as well as the distance from the cylinder.

2. How is the potential of a charged cylinder calculated?

The potential of a charged cylinder can be calculated using the formula V = kQ/r, where V is the potential, k is the Coulomb constant, Q is the charge of the cylinder, and r is the distance from the cylinder to the point where the potential is being measured.

3. Is the potential of a charged cylinder affected by the charge distribution along its length?

Yes, the potential of a charged cylinder is affected by the charge distribution along its length. The potential is highest at the ends of the cylinder, where the charge is concentrated, and decreases as you move towards the center of the cylinder.

4. How does the potential of a charged cylinder compare to that of a point charge?

The potential of a charged cylinder is different from that of a point charge. While the potential of a point charge decreases as you move further away, the potential of a charged cylinder can remain constant at certain points depending on the distribution of charge along its length.

5. Can the potential of a charged cylinder be negative?

Yes, the potential of a charged cylinder can be negative. This occurs when the charge is negative and the distance from the cylinder is greater than the charge's size. In this case, the potential is negative because the electric force is attractive and work is required to move a positive charge away from the cylinder.

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