# An electricity and magnetism problem by Sudeep Salgia

Along the axis of a fixed small loop of resistance $$R$$, a short bar magnet with dipole moment $$\vec{M}$$ is moved away from it with a velocity $$v$$. The $$\vec{M}$$ is oriented such that it points away from the ring as shown in the figure. Given that the radius of the loop is $$a$$ and distance of the magnet from the center of the loop is $$x$$, then the magnitude of the force of interaction between the loop and the bar magnet can be writen as $$\displaystyle F = \frac{p}{q} . \frac{\mu _0^r M^s a^t v^j}{Rx^k}$$ where $$j,k,p,q,r,s,t$$ are natural numbers with $$p$$ and $$q$$ coprime.
Find the value of $$j+k+p+q+r+s+t-2$$.

Details and Assumptions:

• The bar magnet is short and $$a << x$$.

• $$\mu _0$$ is the permeability of vacuum.

• The magnetic field due to the magnet can be considered almost parallel to the axis of the loop.

• Magnitude of force of interaction between two dipoles of moments $$M_1$$ and $$M_2$$ is given by $$\displaystyle F = \frac{6 \mu_0 M_1 M_2 }{4 \pi x^4}$$ where $$x$$ is the distance between them.

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