A length L of wire carries a current i. We have to show that if the wire is formed into a circular coil, the maximum torque in a given magnetic field is developed when the coil has one turn only and the maximum torque has the magnitude

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Let us assume that wire of length L is turned into a coil of radius r and n turns. We then have

Total area of the n-turns of the coil will be

It is maximum for . A current carrying loop of area A and current i is effectively a magnetic dipole with dipole moment

,

where is the unit vector to the plane of the loop. Therefore, the dipole moment of loop will be

The torque on a magnetic dipole of moment in magnetic field is

The maximum value of torque is obtained when and are to each other. Its magnitude then is

For our problem

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