The Electric Field — Question 7

Question 7

A vanishing mean field can coexist with a nonzero force fluctuation

A prescribed field switches between $+E_0\hat{\mathbf x}$ and $-E_0\hat{\mathbf x}$ with equal time fractions, where $E_0=200$ N/C. A fixed nonperturbing probe has charge $+3.00$ nC. Ignore switching transients and average over many complete switches.

1. Find the time-averaged field, its root-mean-square magnitude and the mean probe force.

2. Find the root-mean-square force. Explain why applying the mean field does not reproduce the instantaneous mechanical forcing.

Question 7 - Solution

A vanishing mean field can coexist with a nonzero force fluctuation

(a) Method and calculation

$$\langle\mathbf E\rangle=\tfrac12(E_0-E_0)\hat{\mathbf x}=\mathbf0.$$$$E_{\mathrm{rms}}=\sqrt{\langle E_x^2\rangle}=E_0=200\,\mathrm{N/C},\quad\langle\mathbf F\rangle=q\langle\mathbf E\rangle=\mathbf0.$$

(b) Interpretation and further calculation

$$F_{\mathrm{rms}}=|q|E_{\mathrm{rms}}=6.00\times10^{-7}\,\mathrm N.$$$$\langle E_x^2\rangle=40000\,\mathrm{(N/C)^2}\ne\langle E_x\rangle^2.$$A zero first moment does not remove fluctuations; motion can depend on the switching time scale.

Physical check

This is a time-dependent prescribed-field model, not a static field-line picture.