Proportionality of Induced Current in Zero Resistance Loop

To determine how the induced current in the zero-resistance loop depends on the magnetic dipole moment $m$ and the axial distance $r$, we apply the principle of conservation of magnetic flux for a superconducting (zero-resistance) circuit.

1. Conservation of Magnetic Flux

Since the loop has zero resistance, any change in external magnetic flux creates an induced current that generates an equal and opposite self-flux to keep the total magnetic flux through the loop constant: $\Phi_{\text{total}} = \Phi_{\text{ext}} + \Phi_{\text{self}} = \text{constant}$

Initially, when the dipole $m$ is at infinity ($r \to \infty$), the flux through the loop is zero. Therefore, at any distance $r$, the total flux must remain zero: $\Phi_{\text{ext}} + \Phi_{\text{self}} = 0 \implies |\Phi_{\text{self}}| = |\Phi_{\text{ext}}|$

2. External Magnetic Flux ($\Phi_{\text{ext}}$)

The magnetic field along the axis of a magnetic dipole of moment $m$ at a distance $r$ ($r \gg a$) is given as: $B(r) = \frac{\mu_0 m}{2\pi r^3}$

Since the radius of the loop $a \ll r$, the magnetic field $B(r)$ across the area of the loop $A = \pi a^2$ can be considered uniform. Thus, the external magnetic flux through the loop is: $\Phi_{\text{ext}} = B(r) \cdot A = \left(\frac{\mu_0 m}{2\pi r^3}\right) (\pi a^2) = \frac{\mu_0 m a^2}{2 r^3}$

3. Self-Flux ($\Phi_{\text{self}}$) and Induced Current ($I$)

The flux produced by the induced current $I$ through the loop is given by: $\Phi_{\text{self}} = L I$ where $L$ is the self-inductance of the loop, which depends only on its fixed geometry.

Equating the magnitudes of self-flux and external flux: $L I = \frac{\mu_0 m a^2}{2 r^3}$

Solving for the induced current $I$: $I = \left(\frac{\mu_0 a^2}{2 L}\right) \frac{m}{r^3}$

Since $\mu_0$, $a$, and $L$ are constants: $I \propto \frac{m}{r^3}$

Thus, the induced current in the loop is proportional to $\dfrac{m}{r^3}$.

Correct Option: (A)