The Electric Field — Question 1

Question 1

The weak-probe limit removes the probe-induced disturbance

A small positive probe perturbs a source. Measurements of $F_x/q$ give 120 N/C at $q=1.0$ nC and 140 N/C at $q=2.0$ nC. Assume $F_x/q=E_0+\beta q$, with $q$ expressed in nC. Define the undisturbed field by $E_0=\lim_{q\to0}F_x/q$.

1. Infer $E_0$ and $\beta$, and find the force on a .500 nC probe in this model.

2. Find the largest positive probe charge for which using $F_x/q$ as $E_0$ introduces at most 1 percent relative error.

Question 1 - Solution

The weak-probe limit removes the probe-induced disturbance

(a) Method and calculation

$$120=E_0+\beta(1),\quad140=E_0+\beta(2)\Rightarrow\beta=20\,\mathrm{(N/C)/nC}.$$$$E_0=100\,\mathrm{N/C},\quad F_x(.500)=(.500\times10^{-9})(110)=5.50\times10^{-8}\,\mathrm N.$$

(b) Interpretation and further calculation

$$\frac{|F_x/q-E_0|}{E_0}=\frac{20q}{100}\le.01.$$$$0<q\le.0500\,\mathrm{nC}.$$The field definition requires an arbitrarily weak test charge when the source rearranges in response to the probe.

Physical check

The probe force tends to zero as $q\to0$, but the ratio approaches a finite field.