Conductors and Insulators — Question 9

Question 9

Electroscope leaves can collapse and then reopen without a change of net charge

An isolated electroscope has net charge $+6.0$ nC. A supplied redistribution model assigns each leaf charge $q_L=(3.0-x)$ nC and the knob the remaining charge, where $x$ increases as an external rod approaches without contact. At a fixed leaf geometry, the repulsion magnitude is $F=Cq_L^2$, with constant $C>0$.

1. Find knob charge as a function of $x$ and the value of $x$ at which the leaf repulsion vanishes.

2. At $x=5.0$, find leaf and knob charges and the force ratio to $x=0$. Explain why renewed divergence does not by itself prove that the electroscope net charge changed sign.

Question 9 - Solution

Electroscope leaves can collapse and then reopen without a change of net charge

(a) Method and calculation

$$Q=2q_L+q_K=6.0\,\mathrm{nC}\Rightarrow q_K=2x\,\mathrm{nC}.$$$$F=0\Rightarrow q_L=0\Rightarrow x=3.0.$$

(b) Interpretation and further calculation

$$x=5.0:\quad q_L=-2.0\,\mathrm{nC},\quad q_K=+10.0\,\mathrm{nC},\quad2q_L+q_K=+6.0\,\mathrm{nC}.$$$$F(5)/F(0)=(-2)^2/3^2=4/9.$$The leaves repel for either common sign. Redistribution can reverse their individual signs while the electroscope remains positive overall.

Physical check

The force ratio is evaluated at fixed geometry; actual leaf angles also respond mechanically to the changing force.