Conductors and Insulators — Question 2

Question 2

Grounding order determines whether induction leaves a lasting charge

A neutral metal sphere is held near a positive rod without contact. While the rod stays in place, grounding brings electrons of total charge $-5.0$ nC from Earth to the sphere. Compare sequence I: disconnect ground, then remove rod; and sequence II: remove rod while still grounded, allow equilibrium, then disconnect. In sequence II assume the isolated surroundings are otherwise free of charges, so the grounded sphere returns to neutrality.

1. Find the final sphere and Earth charge changes in sequence I.

2. Find the additional charge transfer during sequence II and the final charges. Explain why the external rod need not supply the electrons.

Question 2 - Solution

Grounding order determines whether induction leaves a lasting charge

(a) Method and calculation

$$Q_{s,1}=-5.0\,\mathrm{nC},\quad\Delta Q_{E,1}=+5.0\,\mathrm{nC}.$$$$\text{ground removed first}:\quad Q_{s,f}=-5.0\,\mathrm{nC}.$$Removing the rod changes the surface distribution but cannot remove the isolated sphere charge.

(b) Interpretation and further calculation

$$\text{rod removed first}:\quad\Delta Q_{s,2}=+5.0\,\mathrm{nC},\quad Q_{s,f}=0.$$$$\Delta Q_{E,\mathrm{total}}=+5.0-5.0=0.$$Electrons return to Earth through the remaining ground connection. The rod controls redistribution and exchange with Earth without touching the sphere.

Physical check

The sphere is an open charge system during grounding and a closed charge system after the wire is removed.