# Mutual Inductance Between a Coil and a Solenoid

1. The problem statement, all variables and given/known data

A small, thin coil with N2 loops, each of area A2, is placed inside a long solenoid, near its center. The solenoid has N1 loops in its length L and has area A1. Find the mutual inductance as a function of θ, the angle between the plane of the small coil and the axis of the solenoid.

2. Relevant equations
M2 = N2$\phi$2/I1
$\phi$2 = BA2cos(θ) = μ0(N1/L)I1A2cos(θ)
3. The attempt at a solution
If we just substitute for $\phi$2 into the equation for M2, we get that
M2 = (N2/I1)BA2cos(θ) = μ0(N1N2/L)A2cos(θ)

Everything is right here except that the correct solution has sin(θ) instead of cos(θ). Why is that? Isn’t the magnetic flux defined as a dot product?

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