Electromagnetic repulsion force

[erisedk]

Homework Statement

An infinitesimally small bar magnet of dipole moment $\vec{M}$ is pointing and moving with the speed v in the x direction. A small closed circular conducting loop of radius a and negligible self inductance lies in the y-z plane with its centre at x = 0, and its axis coinciding with the x axis. Find the force opposing the motion of the magnet, if the resistance of the loop is R. Assume that the distance x of the magnet from the centre of the loop is much greater than a.

Homework Equations

The Attempt at a Solution

Magnetic field due to the magnet of dipole moment $\vec{M}$ on its axis at a distance x (i.e. at the centre of the ring) = $B = \dfrac{2μ_0M}{4πx^3} = \dfrac{μ_0M}{2πx^3}$

$φ_{ring} = B.A = \dfrac{μ_0M}{2πx^3}.πa^2 = \dfrac{μ_0Ma^2}{2x^3}$

$|ε| = \dfrac{dφ}{dt} = \dfrac{3μ_0Ma^2v}{2x^4}$

$i$ (flowing in ring) = $\dfrac{3μ_0Ma^2v}{2x^4R}$

After this, I'm confused.
My attempt:
$F = ilB = \dfrac{3μ_0Ma^2v}{2x^4R}.2a.\dfrac{μ_0M}{2πx^3}$

$F = \dfrac{3μ_0M^2a^3v}{2x^7Rπ}$

This answer is wrong. It is surely due to me having taken l = 2a. Usually, when we consider arbitrarily shaped conductors in a uniform magnetic field (it is uniform on the ring as x>>a), the force, i.e.
$\vec{F} = \int I \vec{dL} × \vec{B} = I \vec{L} × \vec{B}$ where $\vec{L}$ is the length vector joining initial and final points of the conductor. Now, in case of a circular loop, this should've been zero. But that is clearly wrong as there is some force due to lenz's law. I believe it is wrong because the lorentz force equation can't be applied to currents induced due to the magnetic field. It can only be used when we have a current carrying conductor placed in an external magnetic field (I think). In any case, I'm not sure what to do. Please help.

 

[TSny]

I think your expression for the current looks good. If you approximate the magnetic field as purely in the x direction at the loop, would there be any net magnetic force on the loop? Consider the direction of the magnetic force on a small element of length of the loop if the B field has only an x component.

 

[erisedk]

If you approximate the magnetic field as purely in the x direction at the loop, would there be any net magnetic force on the loop?

Zero, because magnetic force on a coil of wire is always zero in a uniform magnetic field?

 

[TSny]

Yes. For the force calculation, you'll need to take into account that B is not actually uniform.

 

[erisedk]

How do I start?

 

[TSny]

Consider the component of B that is perpendicular to the x axis.
https://en.wikipedia.org/wiki/Magnetic_dipole