# Electromagnetic induction

## Homework Statement

A uniform conducting ring of mass π kg and radius 1 m is kept on a smooth horizontal table. A uniform but time varying magnetic field ##\vec{B} = (\hat{i} + t^2\hat{j} ) T## is present in the region (where t is in sec and the positive y-axis is in vertically upward direction, g = 10 m/s2). Resistance of the ring is 2 Ω.

1. Time at which the ring will start toppling is
(A) 10/π sec
(B) 20/π sec
(C) 5/π sec
(D) 25/π sec

2. Heat generated in the ring till the instant when ring starts toppling is
(A) 1/3π kJ
(B) 2/π kJ
(C) 2/3π kJ
(D) 1/π kJ

## Homework Equations

ε = -dφ/dt
i = ε/R
φ = ##\vec{B}.\vec{A}##

## The Attempt at a Solution

I only know that a current will be induced due to the changing magnetic field in the loop and it'll be given by:
##ε = \dfrac{d(\hat{i} + t^2\hat{j} ).(π.1^2\hat{j})}{dt}## and i = ε/R
ie. i = πt
I don't know how this ring will topple. The forces involved will be the magnetic force and the weight of the ring. I don't understand how to write the torque equation for toppling as I don't get how the induced current is responsible for making the ring topple.

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I don't know how this ring will topple. The forces involved will be the magnetic force and the weight of the ring. I don't understand how to write the torque equation for toppling as I don't get how the induced current is responsible for making the ring topple.

as the B field is varying with time induced emf and current will be produced in the ring -by Lenz's law the direction of the induced current will be such that it will oppose the cause and attractive force will be generated which ultimately topple the ring.
you have to draw how the attraction takes place based on Faraday law of electromagnetic induction.

Ok, so the force acting on the ring will be i(l × B) i.e π.t.(2r.t2) (Taking l = 2r). Won't this force act at the centre? Why should it act somewhere else, which I assume it does, because if both gravity and magnetic force act at the centre the ring will never topple.

Ok, so the force acting on the ring will be i(l × B) i.e π.t.(2r.t2) (Taking l⊥ = 2r). Won't this force act at the centre? Why should it act somewhere else, which I assume it does, because if both gravity and magnetic force act at the centre the ring will never topple.

the magnetic field lines couple with the conductor and the time rate of change of flux crossing leads to an induced emf in the body of the ring and the direction of current in the ring will be producing a field due to this induced current will have its direction different w.r.to the external field in the sense that where-in the region the field is increasing the induced field will be opposing this effect - the direction of the induced field will be different in the two halves of the ring -a visualization has to be drawn so that the ring is toppled.
the current will produce heat-thermal change.
the details have to be calculated.

the direction of the induced field will be different in the two halves of the ring
I'm sorry, I really can't visualise this.

Funny thing is, though I can't visualise it, I can calculate the answer now.
Torque due to magnetic field = m×B= iAB = mgr (where B is in x direction as that will be perpendicular to the magnetic moment vector)
πt × π × 12 × 1 = π × 10 × 1
t = 10/π sec

Heat generated will be the integral of i2Rdt from 0 to 10/π = 2/3π kJ

Perhaps I'll go read more on torques due to magnetic fields. Maybe I'll be able to visualise things better then. Thank you so much!

Edit: Got it completely now.