Electric Current — Question 4

Question 4

A changing branch current requires temporary charge storage

For $0\le t\le T$, currents $I_1=I_0$ and $I_2=I_0t/T$ enter a small node, while $I_3=3I_0t/T$ leaves it. Initially its excess charge is zero.

1. Derive the stored charge and determine its maximum.

2. Find the final charge and explain whether zero final charge permits unequal instantaneous currents.

Question 4 - Solution

A changing branch current requires temporary charge storage

(a) Method and calculation

$$dQ/dt=I_1+I_2-I_3=I_0(1-2t/T).$$$$Q(t)=I_0(t-t^2/T),\quad Q_{\max}=I_0T/4\ \text{at }t=T/2.$$

(b) Interpretation and further calculation

$$Q(T)=I_0(T-T)=0.$$$$\int_0^T(I_{\mathrm{in}}-I_{\mathrm{out}})\,dt=0,\quad I_{\mathrm{in}}\ne I_{\mathrm{out}}\ \text{in general}.$$

Physical check

The usual equal-current junction rule is the special case of negligible charge accumulation.