Electric Current — Question 2

Question 2

A cumulative charge curve reveals an instantaneous reversal

The signed charge crossing a plane is $Q(t)=at^2-bt^3$ for $0\le t\le a/b$, where $a,b>0$ have units C/s$^2$ and C/s$^3$.

1. Find the current and its nonzero reversal time.

2. Find the net charge and the total magnitude of charge transported over this interval.

Question 2 - Solution

A cumulative charge curve reveals an instantaneous reversal

(a) Method and calculation

$$I=dQ/dt=2at-3bt^2=t(2a-3bt).$$$$t_*=2a/(3b),\quad Q(t_*)=4a^3/(27b^2).$$

(b) Interpretation and further calculation

$$Q(a/b)-Q(0)=0.$$$$\int_0^{a/b}|I|\,dt=2Q(t_*)=8a^3/(27b^2).$$

Physical check

Zero net transport does not imply that no charge crossed the plane.