# Mutual Inductance Between a Coil and a Solenoid

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

A small, thin coil with N_{2} loops, each of area A_{2}, is placed inside a long solenoid, near its center. The solenoid has N_{1} loops in its length L and has area A_{1}. 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**

M_{2} = N_{2}[itex]\phi[/itex]_{2}/I_{1}

[itex]\phi[/itex]_{2} = BA_{2}cos(θ) = μ_{0}(N_{1}/L)I_{1}A_{2}cos(θ)

**3. The attempt at a solution**

If we just substitute for [itex]\phi[/itex]_{2} into the equation for M_{2}, we get that

M_{2} = (N_{2}/I_{1})BA_{2}cos(θ) = μ_{0}(N_{1}N_{2}/L)A_{2}cos(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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