Energy Stored in an Electric Field — Question 5

Question 5

Integrating coaxial field energy reproduces the terminal formula

A long vacuum coaxial capacitor has radii a,b and length L, with opposite line charges $\pm\lambda$. Use $E(r)=\lambda/(2\pi\epsilon_0r)$ and energy density $u=\epsilon_0E^2/2$.

1. Integrate the energy throughout the gap.

2. Express the result as Q squared over 2C and find what fraction lies between a and $\sqrt{ab}$.

Question 5 - Solution

Integrating coaxial field energy reproduces the terminal formula

(a) Method and calculation

$$U=\int_a^b\frac{\epsilon_0E^2}{2}(2\pi rL)dr=\frac{\lambda^2L}{4\pi\epsilon_0}\int_a^b\frac{dr}{r}.$$$$U=\frac{\lambda^2L}{4\pi\epsilon_0}\ln(b/a).$$

(b) Interpretation and further calculation

$$Q=\lambda L,\quad C=\frac{2\pi\epsilon_0L}{\ln(b/a)}\Rightarrow U=Q^2/(2C).$$$$\frac{U(a<r<\sqrt{ab})}{U}=\frac{\ln(\sqrt{ab}/a)}{\ln(b/a)}=\frac12.$$

Physical check

Equal logarithmic radial intervals contain equal energy in this geometry, despite their unequal physical volumes.