Mutual Inductance Between a Coil and a Solenoid

[asap9993]

Homework Statement

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.

Homework Equations

M₂ = N₂$\phi$₂/I₁
$\phi$₂ = BA₂cos(θ) = μ₀(N₁/L)I₁A₂cos(θ)

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?

 

[collinsmark]

Homework Statement

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.

Homework Equations

M₂ = N₂$\phi$₂/I₁
$\phi$₂ = BA₂cos(θ) = μ₀(N₁/L)I₁A₂cos(θ)

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?

I think it has to do with the wording, "the angle between the plane of the small coil and the axis of the solenoid."

The axis of the small coil is perpendicular to the plane of the same, small coil.

 

[rude man]

Homework Statement

" ...the angle between the plane of the small coil and the axis of the solenoid. "

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?

You are right. When speaking of the angle a plane makes with another direction, the normal should be understood, although very often it isn't.

A plane has direction only in the sense of its normal.