Mutual Induction — Question 2

Question 2

Changing separation induces a voltage even at constant primary current

Two coaxial coils have positive mutual inductance $M(x)=C/x^3$, where x is separation and C is constant. Coil 2 is open. Define its induced emf as $\mathcal E_2=-d(MI_1)/dt$.

1. Find the emf when both primary current and separation can change.

2. For constant positive I1 and constant outward speed v, find the emf and the current law that would instead cancel it.

Question 2 - Solution

Changing separation induces a voltage even at constant primary current

(a) Method and calculation

$$\mathcal E_2=-M\dot I_1-I_1\frac{dM}{dx}\dot x=-\frac{C}{x^3}\dot I_1+\frac{3CI_1\dot x}{x^4}.$$$$\boxed{\mathcal E_2=3CI_1v/x^4\quad(\dot I_1=0)}.$$

(b) Interpretation and further calculation

$$\mathcal E_2=0\Rightarrow d(MI_1)/dt=0\Rightarrow I_1/x^3=\text{constant}.$$$$\boxed{I_1(t)=I_1(0)[x(t)/x(0)]^3}.$$

Physical check

An open secondary has no load current but can still have induced emf. Both current and geometry change the linked flux.