Self-Induction — Question 7

Question 7

Changing inductance can induce an emf even when current is held constant

A linear coil has time-dependent inductance $L(t)=L_0(1+\alpha t)$ because its geometry changes slowly. A current source holds I constant. Ignore winding resistance and hysteresis.

1. Find linkage, terminal voltage and self-induced emf.

2. Find the charge-source electrical work over time T and explain why the fixed-L formula alone is insufficient.

Question 7 - Solution

Changing inductance can induce an emf even when current is held constant

(a) Method and calculation

$$\Lambda=L(t)I\Rightarrow\boxed{v=\dot\Lambda=I\dot L=\alpha L_0I}.$$$$\boxed{\mathcal E_{\mathrm{self}}=-\alpha L_0I}.$$

(b) Interpretation and further calculation

$$W_{\mathrm{elec}}=\int_0^Tvi\,dt=\boxed{\alpha L_0I^2T}.$$$$\frac d{dt}(Li)=L\dot i+i\dot L;\quad\dot i=0\text{ does not remove }i\dot L.$$

Physical check

Mechanical work can also be exchanged when geometry changes; electrical work need not equal the field-energy change alone.