Mutual Induction — Question 1

Question 1

Nested long solenoids share the smaller cross-sectional flux

Two coaxial long vacuum solenoids have the same length ell, turn counts N1 and N2, and cross-sectional areas $A_1<A_2$. Neglect end effects. Their positive currents produce aligned axial fields.

1. Derive M from the flux linkage in coil 2 due to coil 1.

2. Derive M in the reverse direction and explain why the larger area does not replace A1.

Question 1 - Solution

Nested long solenoids share the smaller cross-sectional flux

(a) Method and calculation

$$B_1=\mu_0N_1I_1/\ell,\quad\Lambda_{21}=N_2B_1A_1.$$$$\boxed{M_{21}=\Lambda_{21}/I_1=\mu_0N_1N_2A_1/\ell}.$$

(b) Interpretation and further calculation

$$B_2=\mu_0N_2I_2/\ell,\quad\Lambda_{12}=N_1B_2A_1.$$$$\boxed{M_{12}=M_{21}=M}.$$

Physical check

Only the shared flux area links both coils; reciprocity agrees with the direct geometry calculation.