# Electrostatic potential using method of images

tomwilliam2

## Homework Statement

There is a charge q, at a distance d from an infinite conducting plane (z=0).
Determine the electrostatic potential drop between the z=0 and z=d.

## Homework Equations

Ohm's Law
DeltaV=-integral (E.dl)

## The Attempt at a Solution

I know how to do this problem, using J = sigma E, Ohm's law, model the situation with a charge -q at z=-d, find the electric field, then integrate over the line to find an expression for the potential function V(z). I know I can integrate this potential function over the line now to find the potential drop, using the boundary condition that V=0 on the conducting plane.
The one thing I don't understand is the units. If I have a potential in Volts, and I integrate over a line, how can it result in a potential drop? Wouldn't it be V m?

Homework Helper
Gold Member
Wouldn't the potential drop (or difference) just be V(d)-V(0)? Integrating the electrostatic potential V(z) from z=0 to z=d, would give you the change in electrostatic energy (sometimes confusingly referred to as electrostatic potential energy).

tomwilliam2
I thought that too, but from the worked solutions I get the potential as:

$$V(z) = \frac{I}{4\pi \sigma z} - \frac{I}{4\pi \sigma (2d - z)}$$

Where sigma is conductivity.
It goes on to say: "by integrating this expression between the real source and the metal boundary, we could obtain an expression for the potential drop deltaV, and dividing this by the currnt would give resistance."

I thought it was a mistake, but this online source says the same (para three):
http://physicspages.com/2011/10/08/curl-potential-in-electrostatics/

Is this all just wrong?

Homework Helper
Gold Member
I thought that too, but from the worked solutions I get the potential as:

$$V(z) = \frac{I}{4\pi \sigma z} - \frac{I}{4\pi \sigma (2d - z)}$$

Where sigma is conductivity.
It goes on to say: "by integrating this expression between the real source and the metal boundary, we could obtain an expression for the potential drop deltaV, and dividing this by the currnt would give resistance."

Looking up worked solutions is not the same as solving a problem yourself. You'll get much more out of doing the latter.

In any case, the beginning of the statement "by integrating this expression" is ambiguous without context that allows one to determine which expression they are talking about integrating. I suspect it is the electric field they are integrating to get the potential drop or difference.

I thought it was a mistake, but this online source says the same (para three):
http://physicspages.com/2011/10/08/curl-potential-in-electrostatics/

Is this all just wrong?

Here they are definitely talking about integrating the electric field, not the potential. This should be clear from the equation in the middle of the paragraph.

$$V(b) - V(a) = -\int_a^b \mathbf{E} \cdot d\mathbf{l}$$

tomwilliam2
Thanks for the help.
I think my textbook has a mistake in it...that's why when it uses the final expression to calculate resistance it comes out in Ohm metres.

It was worth going through the worked example with a fine toothcomb after all!