Electric Current — Question 7

Question 7

Discrete charge packets create a mean current with counting fluctuations

Use $e=1.602176634\times10^{-19}$ C, exact. Each detected packet transports $2e$ toward $+x$. In a gate of duration T the count N has supplied statistics $\langle N\rangle=\lambda T$ and $\operatorname{Var}N=\lambda T$. Packets are independent.

1. Find the mean and standard deviation of the gate-averaged current.

2. How long must the gate be for fractional standard deviation at most 1.00 percent if $\lambda=4.0\times10^6\,\mathrm{s^{-1}}$?

Question 7 - Solution

Discrete charge packets create a mean current with counting fluctuations

(a) Method and calculation

$$I_T=2eN/T,\quad\langle I_T\rangle=2e\lambda.$$$$\sigma_I=(2e/T)\sqrt{\lambda T}=2e\sqrt{\lambda/T}.$$

(b) Interpretation and further calculation

$$\sigma_I/\langle I_T\rangle=1/\sqrt{\lambda T}\le .0100.$$$$T\ge10^4/\lambda=2.50\,\mathrm{ms}.$$

Physical check

Counting fluctuations shrink as the square root of gate duration, even though the mean current stays constant.