Question

Spinning a loop of wire between the poles of a magnet will induce an electric current. Which of the following conditions will minimize the induced current?
1. The plane of the loop is perpendicular to the magnetic field
2. The plane of the loop is halfway between parallel and perpendicular
3. The plane of the loop is parallel to the magnetic field
4. The magnetic flux through the wire is minimized

Answer 1

Faraday's Law is

\(\mathcal{E} = - \dfrac{d\Phi }{dt}\)

Magnetic flux (\(\Phi\)) through a surface is the sum of all the the normal components of the magnetic field \(\mathbf B \) passing through that surface.

Here with uniform magnetic field \(\mathbf B \), the flux is \(\Phi = BA \cos \theta\).

Thusly

\( \dfrac{d\Phi }{dt} = - B A \sin \theta \ \dot \theta\)

And so:

\(\mathcal E = i R = B A \sin \theta \ \dot \theta\)

All other things being equal:

\(i \propto \sin \theta \) and \(|i| \propto \sin \theta \)

Looking at the options:

1. The plane of the loop is perpendicular to the magnetic field, \(\implies \theta = 0, \pi\)

2. The plane of the loop is halfway between parallel and perpendicular \(\implies \theta = {\pi \over 4}, {5\pi \over 4}\)

3. The plane of the loop is parallel to the magnetic field \(\implies \theta = {\pi \over 2}, {3\pi \over 2}\)

4. The magnetic flux through the wire is minimized \(\implies \theta = {\pi \over 2}, {3\pi \over 2}\)


So your min should be in option 1 provided we are talking about absolute values of current, which makes sense. All that happens is that the current changes direction for negative values of \(\sin \theta\).