Potential Due to a Point Charge — Question 7

Question 7

Radial uncertainty produces asymmetric exact voltage bounds

A positive point charge has exact kQ=12.0 V m. The observation radius lies within $r=.300\pm.030$ m as a hard interval. Use zero potential at infinity.

1. Find the nominal voltage and exact allowed voltage interval.

2. Compare the exact deviations with the first-order estimate $|\delta V|/V\simeq|\delta r|/r$.

Question 7 - Solution

Radial uncertainty produces asymmetric exact voltage bounds

(a) Method and calculation

$$V_{\mathrm{nominal}}=12.0/.300=40.0\,\mathrm V.$$$$V\in[12.0/.330,12.0/.270]=[36.3636,44.4444]\,\mathrm V.$$

(b) Interpretation and further calculation

$$\delta V_{\mathrm{low}}=-3.63636\,\mathrm V,\quad\delta V_{\mathrm{high}}=+4.44444\,\mathrm V.$$$$|\delta V|_{\mathrm{linear}}\simeq40.0(.030/.300)=4.00\,\mathrm V.$$

Physical check

The inverse-radius function is convex, so a symmetric radius interval does not produce a symmetric voltage interval.