"Pumping" Charges — Question 6

Question 6

Unequal entrance and exit currents would charge the source as a whole

A source enclosure initially has zero net charge. A current 2.00 A enters its negative terminal while 1.97 A leaves its positive terminal, for 0.400 s. No other current crosses the enclosure.

1. Find the net accumulated charge and its sign.

2. If the enclosure has capacitance 50.0 pF to distant ground, estimate its common-potential shift and explain why such a mismatch cannot represent ordinary steady operation.

Question 6 - Solution

Unequal entrance and exit currents would charge the source as a whole

(a) Method and calculation

$$\frac{dQ_{\mathrm{box}}}{dt}=I_{\mathrm{in}}-I_{\mathrm{out}}=.0300\,\mathrm A.$$$$\Delta Q_{\mathrm{box}}=(.0300)(.400)=.0120\,\mathrm C>0.$$

(b) Interpretation and further calculation

$$\Delta V_{\mathrm{common}}=\Delta Q_{\mathrm{box}}/C_g=.0120/(50.0\times10^{-12})=2.40\times10^8\,\mathrm V.$$$$\text{Steady state: }dQ_{\mathrm{box}}/dt=0\Rightarrow I_{\mathrm{in}}=I_{\mathrm{out}}.$$

Physical check

The enormous idealized voltage exposes the inconsistency; neglected leakage and breakdown would intervene long before it could occur.