Charge Is Conserved — Question 7

Question 7

A decaying Coulomb force does not imply destruction of electric charge

Two identical positive pointlike spheres remain a fixed distance apart. Each leaks charge to a common environment according to $dQ/dt=-Q/\tau$, with initial charge $Q_0$ and $\tau=12.0$ s. Use Coulomb law for their mutual force; other effects of the environment are neglected in this specified model.

1. Find the sphere charge and mutual force as functions of time. Determine when the force first falls to half its initial value.

2. At that time find the total charge gained by the environment, which began neutral. Verify the complete-system charge balance.

Question 7 - Solution

A decaying Coulomb force does not imply destruction of electric charge

(a) Method and calculation

$$Q(t)=Q_0e^{-t/\tau},\quad F(t)=\frac{kQ(t)^2}{r^2}=F_0e^{-2t/\tau}.$$$$F(t_{1/2})=F_0/2\Rightarrow t_{1/2}=\frac\tau2\ln2\approx4.15888\,\mathrm s.$$

(b) Interpretation and further calculation

$$Q(t_{1/2})=Q_0/\sqrt2,\quad Q_{\mathrm{env}}=2Q_0(1-1/\sqrt2).$$$$2Q(t_{1/2})+Q_{\mathrm{env}}=2Q_0.$$The electrical interaction weakens because the charges leave the spheres and enter the environment, not because total electric charge ceases to exist.

Physical check

The factor of two in the force decay exponent follows because both source charges decay.