# Mutual Inductance Between a Coil and a Solenoid

## 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 Helper
Gold Member

## 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 Helper
Gold Member

## 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.