Mutual Induction — Question 8

Question 8

Fixed-current mutual energy requires accounting for both sources to predict force

Two coaxial coils have $M(x)=C/x^3>0$ and constant self-inductances. Sources hold positive currents I1 and I2 fixed while separation x changes quasistatically. Ignore resistance. Positive force points toward increasing x.

1. Find electrical work and field-energy change for a small dM.

2. Use energy balance to obtain mechanical force and its direction.

Question 8 - Solution

Fixed-current mutual energy requires accounting for both sources to predict force

(a) Method and calculation

$$dW_e=I_1d(MI_2)+I_2d(MI_1)=2I_1I_2dM.$$$$dU=d(MI_1I_2)=I_1I_2dM\Rightarrow dW_{\mathrm{mech}}=dW_e-dU=I_1I_2dM.$$

(b) Interpretation and further calculation

$$F_xdx=dW_{\mathrm{mech}}\Rightarrow F_x=I_1I_2\frac{dM}{dx}.$$$$\boxed{F_x=-3CI_1I_2/x^4<0\quad\text{(attraction)}}.$$

Physical check

Using minus the fixed-current field-energy gradient alone would reverse the force because it omits work from the current sources.