Induced Electric Fields — Question 6

Question 6

A fixed-radius induction accelerator requires a relation between average and orbit fields

A particle circulates at fixed radius r in an axially symmetric field. Let Bavg be the signed area-average field inside the orbit and Borb the field at the orbit. Use magnitudes in an accelerating interval: $p=|q|rB_{\mathrm{orb}}$ and $\dot p=|q||E_\phi|$, with both relevant field magnitudes increasing.

1. Derive the relation between their time derivatives required for fixed r.

2. Integrate it and state when the familiar factor-of-two field relation follows.

Question 6 - Solution

A fixed-radius induction accelerator requires a relation between average and orbit fields

(a) Method and calculation

$$2\pi r|E_\phi|=\pi r^2\dot B_{\mathrm{avg}}\Rightarrow|E_\phi|=r\dot B_{\mathrm{avg}}/2.$$$$\dot p=|q|r\dot B_{\mathrm{orb}}=|q|r\dot B_{\mathrm{avg}}/2\Rightarrow\boxed{\dot B_{\mathrm{avg}}=2\dot B_{\mathrm{orb}}}.$$

(b) Interpretation and further calculation

$$\boxed{B_{\mathrm{avg}}(t)-2B_{\mathrm{orb}}(t)=\text{constant}}.$$$$B_{\mathrm{avg}}(0)=2B_{\mathrm{orb}}(0)\Rightarrow\boxed{B_{\mathrm{avg}}(t)=2B_{\mathrm{orb}}(t)}.$$

Physical check

The derivative condition alone does not set the integration constant. The initial orbit and field configuration must also be compatible.