The Electric Field Due to a Point Charge — Question 7

Question 7

A nonradial residual rejects the hypothesis of a single source at the origin

Use $k=8.99\times10^9\,\mathrm{N\,m^2/C^2}$; treat the specified charges as points in vacuum. At P$=(.300,.400,0)$ m, a measured field is $(6,8,2)$ N/C. A proposed model places one point charge at the origin and no other sources.

1. Test the direction predicted by that model and find the charge that reproduces the radial projection of the measurement.

2. Find the residual field not explained by that charge and its fraction of the measured magnitude.

Question 7 - Solution

A nonradial residual rejects the hypothesis of a single source at the origin

(a) Method and calculation

$$r=.500\,\mathrm m,\quad\hat{\mathbf r}=(.6,.8,0),\quad E_r=\mathbf E\cdot\hat{\mathbf r}=10.0\,\mathrm{N/C}.$$$$Q=E_rr^2/k\approx2.78087\times10^{-10}\,\mathrm C,\quad\mathbf E_{\mathrm{model}}=(6,8,0)\,\mathrm{N/C}.$$

(b) Interpretation and further calculation

$$\mathbf E_{\mathrm{residual}}=(0,0,2)\,\mathrm{N/C},\quad\frac{|\mathbf E_{\mathrm{residual}}|}{|\mathbf E|}=\frac2{\sqrt{104}}\approx.196116.$$$$\mathbf E_{\mathrm{measured}}\not\parallel\mathbf r.$$A charge adjustment can change radial magnitude but cannot create the required out-of-plane component.

Physical check

The residual identifies failure of the stipulated one-source model, not the location of any omitted source.