Problem — Electrostatic induction on a metal sphere

Problem

A positively charged body is brought close to a neutral metal sphere resting on an insulating pedestal. Explain how the sphere is attracted while remaining globally neutral.

Solution

Electrostatic induction. In the metal, conduction electrons are free to move. The field of the positive body $+Q$ attracts them: electrons crowd onto the face of the sphere nearer to $+Q$, leaving the far face with a deficit of electrons, i.e. positive charge. Two zones of induced charge thus appear on the sphere, $-$ near and $+$ far.

The sphere stays neutral. No charge has entered or left the sphere (it is isolated by the pedestal): the negative charge crowded on one side is exactly compensated by the positive charge on the other. The total charge remains zero:

$q_{\text{tot}} = q_{-} + q_{+} = 0$

Why there is a net attraction. The Coulomb force depends on distance as $1/r^2$. The induced negative charges are closer to $+Q$ than the induced positive charges, which lie on the opposite face. Hence:

$F_{\text{attr}}\propto \frac{q_{-}}{r_{\text{near}}^2} \;>\; F_{\text{rep}}\propto \frac{q_{+}}{r_{\text{far}}^2}$

The attraction on the nearby charges outweighs the repulsion on the far charges, and the net result is a net attraction towards $+Q$, even though the sphere is globally neutral.

$\ev{\text{neutral sphere: net attraction because the induced }-\text{ charges are closer to }+Q}$

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Topics: Electrostatics Concepts: Electric charge · Conductors in electrostatic equilibrium