Charge Is Conserved — Question 2

Question 2

Negative charge leaving a region increases its net charge

A detector initially has charge $-2.0$ nC. Positive particles enter carrying $+6.0$ nC/s, electrons enter carrying $-4.0$ nC/s, and electrons leave carrying $-1.0$ nC/s. All three signed charge rates remain constant. Ignore every other transfer.

1. Find the detector charge accumulation rate and its charge after .500 s.

2. Find when the detector first becomes neutral and the corresponding net charge change of the external environment.

Question 2 - Solution

Negative charge leaving a region increases its net charge

(a) Method and calculation

$$\frac{dQ_D}{dt}=\dot Q_{\mathrm{in}}-\dot Q_{\mathrm{out}}=(6.0-4.0)-(-1.0)=+3.0\,\mathrm{nC/s}.$$$$Q_D(t)=-2.0+3.0t\quad\mathrm{nC}\quad(t\text{ in seconds}),\quad Q_D(.500)=-.500\,\mathrm{nC}.$$

(b) Interpretation and further calculation

$$Q_D(t_0)=0\Rightarrow t_0=2/3\,\mathrm s.$$$$\Delta Q_D=+2.0\,\mathrm{nC},\quad\Delta Q_{\mathrm{environment}}=-2.0\,\mathrm{nC}.$$Subtracting an outward negative charge rate gives a positive contribution to accumulation.

Physical check

The sign of a carrier and whether it crosses inward or outward are separate pieces of the balance.