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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