Squares, Circles and... Magnetic Fields?

Electricity and Magnetism Level 2

A square loop and a circular loop made up of the same metal wires are moving out of a uniform magnetic field region to a field-free region with the same constant velocity.

In which loop will the induced EMF be constant during the passage through the field region?

The magnetic field is normal to both loops.

Square loop Circle loop It's the same in both loops

Abha Vishwakarma by Abha Vishwakarma , 20, India

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1 solution

Abha Vishwakarma
Oct 12, 2018

The EMF induced in a loop is equal to the rate of change of magnetic flux through it.

E = − d Φ d t \large{\mathcal{E} = - \frac{d \Phi}{d t}}

The magnetic flux through an area A A is defined as

Φ = B ⋅ A . \Phi = \mathbf{B} \cdot \mathbf{A}.

It is given that the magnetic field is uniform, so the only factor that changes the magnetic flux when the loops move to the field free region is the area A A (note that is not the area of the loop. It is the area inside the loop that lies in the magnetic field region).

As the angle between the area vector and magnetic field vector is 0 0 in this situation, the emf induced is given by

E = − B d A d t \large{\mathcal{E} = - B\frac{dA}{d t}}

For the emf to remain constant, the rate of change of flux should be constant. So for this particular case, the rate of change of area should remain constant.

Only in the case of the square loop is d A d t \large\frac{dA}{dt} constant. Hence emf is constant in the square loop.

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