Physics Arena

Electrostatics Intermediate metal-in-electric-field

Source: High school physics (Chinese)

Problem Sets:

electrostatics 2

Problem

A conducting rod of length $l$ is initially uncharged. A point charge of magnitude $q$ is placed on the axis of the rod, at a distance $r$ from the rod's left end. After electrostatic equilibrium is reached, induced charges appear on the rod.

What is the magnitude of the electric field at the midpoint of the rod produced by the induced charges alone?

$E_{\text{induced}} = \dfrac{4kq}{(2r + l)^2}$ (equivalently, $\dfrac{kq}{(r + l/2)^2}$).

At every interior point of a conductor in electrostatic equilibrium, the total electric field must be zero. The total field is the superposition of (i) the field from the external point charge $q$ and (ii) the field from the induced charges on the rod. Hence at the midpoint of the rod the two contributions cancel:

$$\mathbf{E}_{\text{induced}} = -\mathbf{E}_q.$$

The midpoint of the rod is at distance $r + l/2$ from the external charge, so the field from $q$ at that point has magnitude

$$E_q = \dfrac{kq}{(r + l/2)^2} = \dfrac{4kq}{(2r + l)^2}.$$

The induced-charge field has the same magnitude (and is directed oppositely along the axis of the rod):

$$E_{\text{induced}} = \dfrac{4kq}{(2r + l)^2}.$$