Induced Magnetic Fields — Question 7

Question 7

A quasistatic capacitor field has a small magnetic-to-electric energy scale

A uniform axial vacuum field oscillates with amplitude E0 and angular frequency omega in radius a. Its induced magnetic peak at the edge is $B_p=a\omega E_0/(2c^2)$. Compare separately maximized energy densities, not simultaneous values.

1. Find $u_{B,\max}/u_{E,\max}$ using $c^{-2}=\mu_0\epsilon_0$.

2. Evaluate the ratio for $a\omega/c=.0200$ and interpret the quasistatic scale.

Question 7 - Solution

A quasistatic capacitor field has a small magnetic-to-electric energy scale

(a) Method and calculation

$$u_{B,\max}=B_p^2/(2\mu_0),\quad u_{E,\max}=\epsilon_0E_0^2/2.$$$$\boxed{u_{B,\max}/u_{E,\max}=(cB_p/E_0)^2=(a\omega/(2c))^2}.$$

(b) Interpretation and further calculation

$$a\omega/c=.0200\Rightarrow(a\omega/(2c))^2=.000100.$$$$\boxed{u_{B,\max}/u_{E,\max}=10^{-4}}.$$

Physical check

Small size compared with the electromagnetic variation length supports the leading quasistatic approximation.