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Consider a straight segment of a current carrying wire of length L and cross sectional area A. Suppose the amount of current passing through the conductor is I. And the conductor is placed in a uniform magnetic field B. Then the magnetic force acting on the conductor is given by,
This is demonstrated as below.
This can be extended to any arbitrarily shaped wire. For any arbitrary shaped current carrying conductor placed in a magnetic field the corresponding force will be the amount of current times the cross product of integrated length and magnetic field. If there is a closed current loop in a uniform magnetic field then the net magnetic force acting on it is zero.
Torque on a current loop: Consider a current carrying loop enclosing an area A and carrying a current I. If the loop is placed within a magnetic field B, then the torque acting on the loop is given by,
where θ is the angle made by the magnetic field with the normal to the plane of the loop. Thus, we can write the torque as:
The torque is maximum when θ = 90o. Since the product of I and A is the magnetic dipole moment (μ = IA), the torque can now be written as:
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