- #1

- 523

- 0

## Homework Statement

Find the surface charge induced on an infinite conducting plate by a point charge +q at a distance a from the plate.

## The Attempt at a Solution

As you probably know, this problem can be solved using the well-known fact that the induced electric field is equivalent to replacing the conductor by a negative point charge -q placed diametrically oposite to +q. I am trying to solve the problem without using this fact.

Can you spot the error in this logic?

The electric field at a point on the surface due to the point charge +q is given by [itex]\frac{q}{4\pi\epsilon_0 }\frac{a}{r^3}[/itex].

Since the plate is conducting, the surface electric field is [itex]\frac{\sigma}{\epsilon_0}[/itex].

Application of Gauss's law to a pill-box of negligible height and area A, gives

[itex]\oint \vec{E} \cdot d\vec{a} = \frac{Q_\mathrm{encl}}{\epsilon_0}[/itex]

[itex]-\frac{q}{4\pi\epsilon_0}\frac{a}{r^3} A - \frac{\sigma}{\epsilon_0}A = \frac{\sigma}{\epsilon_0}A \implies[/itex]

[itex]\sigma = -\frac{qa}{8\pi\epsilon_0 r^3}[/itex].

Correct answer:

[itex]\sigma = -\frac{qa}{2\pi\epsilon_0 r^3}[/itex].

Edit: Corrected typos.

Last edited: