A small magnet of magnetic moment M is placed at a distance r from the origin O with its axis parallel to x-axis as shown. A small coil of one turn, is placed on the x-axis, at the same distance from the origin, with the axis of the coil coinciding with x-axis. For what value of current in the coil does a small magnetic needle, kept at origin remain undeflected? What is the direction of current in the coil?

Solution

For a needle to remain undeflected the magnetic field due to magnet and current loop must be equal and opposite.

The magnetic field of magnet at origin

= \frac{μ_{0}}{4 π} . \frac{M}{r^{3}}

The magnetic field of the small coil at origin

= \frac{μ_{0}}{4 π} \frac{{Iπx}^{2}}{(r^{2} + x^{2})^{3 / 2}}

where, x is the radius of the loop.

For, the needle to remain undeflected we have

\frac{μ_{0}}{4 π} . \frac{M}{r^{3}} = \frac{μ_{0}}{4 π} \frac{{Iπx}^{2}}{(r^{2} + x^{2})^{3 / 2}}

i.e.,

I = \frac{M {(r^{2} + x^{2})}^{3 / 2}}{{πx}^{2}}

The direction of the current in the loop must be anticlockwise.

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