Problem — Induction on a small metal sphere

Problem

You bring a positively charged rod close (without touching it) to a neutral small metal sphere on an insulating stand. Describe how the conduction electrons redistribute and explain why the sphere is attracted despite remaining overall neutral.

Solution

Charge redistribution (electrostatic induction). In a metal, the conduction electrons are free to move. The positive rod attracts them: the electrons migrate towards the side of the sphere nearest the rod, which thus becomes negatively charged. The far side, deprived of electrons, is left with an excess of positive charge (the lattice ions). The sphere is polarised: a $-$ side near, a $+$ side far.

Total charge unchanged. No charge enters or leaves (the stand is insulating): the sphere remains globally neutral. Only the charge inside it has been separated.

Why it is attracted. The rod exerts an attractive force on the negative (near) region and a repulsive force on the positive (far) region. But the Coulomb force goes as $1/r^2$: the negative region is closer, so the attraction is stronger than the repulsion on the more distant positive region.

$F \propto \frac{1}{r^2},\qquad r_{-} < r_{+} \;\Longrightarrow\; F_{\text{attr}} > F_{\text{rep}}$

The net result is therefore a net attraction.

$\ev{\text{net attraction despite remaining neutral}}$

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Topics: Elettrostatica Concepts: Carica elettrica · Conduttori in equilibrio elettrostatico