# Can emf be induced by reducing the radius of a loop?

Faraday's law tells us, that the induced emf in a conducting loop, is equal to the change in magnetic flux per second through the area encircled by the loop.

If we decrease the radius of the loop, without moving it closer or farther away from the source of a magnetic field, the flux through the loop should also decrease, thus inducing an emf. Is this correct?

• Yes, it is correct. Commented Jan 10, 2019 at 8:16
• Yes. Magentic flux is $\int BdA$ over the area of the loop. EMF will be induced when there is a change to either of the quantities. Commented Jan 10, 2019 at 8:50
• @physicsguy19 But wouldn't that be a case where, contrary to if you would move the coil farther or closer to the magnet, the magnetic field caused by the induced current does not oppose further movement of the coil, and wouldn't that be in violation of energy conservation? Commented Jan 10, 2019 at 10:25

You are correct.

There are a number of ways of looking at the situation.

Assume that there is a magnetic field $$\vec B$$ into the screen and the whole circumference of the loop is moving inwards at a speed $$\vec v$$.

The direction of the induced current can be deduced from the right hand grip rule knowing that Lenz's law predicts that the magnetic field produced by the induced current will be in such a direction as to reduce the change producing it which is a reduction in the magnetic flux through the loop.
The induced magnetic field is into the screen and the induced current is clockwise.

A charge $$q$$ within the loop moving inwards at a velocity $$\vec v$$ experiences a Lorentz force $$q \vec v \times \vec B$$ in a direction as shown in the diagram.
This results in the direction of the induced (conventional - as if the mobile charge carriers were positive) current being clockwise.

Now you have an induced current-carrying conductor moving inward at a speed $$v$$ in a magnetic field $$\vec B$$.
There is therefore a force on the conductor radially outwards.
To maintain the radially inward constant speed motion of the loop an external force needs to be applied radially inwards and this external force does work as the4r loop moves inwards.
The work done by the external force results in the generation of the induced current which when passing through the loop which has resistance produces a heating effect.
This is the law of conservation of energy in action with the work done by the external force ultimately producing heat.