Potential of a Charged Isolated Conductor — Question 5

Question 5

A prescribed surface-field limit sets the largest allowed sphere voltage

Use $k=8.99\times10^9\,\mathrm{N\,m^2/C^2}$. A model requires the field just outside a conducting sphere to remain below $E_*=2.50\times10^6$ V/m. The sphere radius is .0800 m and there are no other sources. Treat the limit as supplied data.

1. Find the limiting charge magnitude and potential magnitude relative to infinity.

2. If the radius doubles at the same field limit, find the factors by which allowable charge and voltage grow.

Question 5 - Solution

A prescribed surface-field limit sets the largest allowed sphere voltage

(a) Method and calculation

$$E_s=k|Q|/R^2\Rightarrow|Q|_{\max}=E_*R^2/k=1.77976\,\mu\mathrm C.$$$$|V|_{\max}=k|Q|_{\max}/R=E_*R=2.00\times10^5\,\mathrm V.$$

(b) Interpretation and further calculation

$$|Q|_{\max}\propto R^2\Rightarrow Q_{\max}(2R)/Q_{\max}(R)=4.$$$$|V|_{\max}\propto R\Rightarrow V_{\max}(2R)/V_{\max}(R)=2.$$

Physical check

These are limits of the stated ideal field model; no universal material breakdown value is assumed.