Question

show that the magnetic dipole moment of any current loop bis equal to the product of current and area of the loop
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Answer 1

Answer:

The magnetic dipole moment of the current loop is equal to the product of ampere-turns and area of current loop.

Answer 2

Answer:

The magnetic dipole moment of any current loop is equal to product of the current and its loop area.

Explanation:

• The magnetic field produced at a large distance r from the centre of a circular loop of radius a along its axis is given by :

$B = \frac{μ₀I {a}^{2} }{2 {r}^{3} } \\ B = \frac{μ₀}{4\pi} \frac{2IA}{ {r}^{3} } - - - (1)$

• where I is the current in the loop and A = πa² is its area.
• On the other hand, the electric field of an electric dipole at the axial point lying far away from it is given by

$E = \frac{1}{4\piε₀} \frac{2p}{ {r}^{3} } - - - (2)$

• where p is the electric dipole moment of the electric dipole.
• On comparing equation (1) and (2), we note that E and B have same distance dependence 1/r³ .
• Moreover, they have same direction at any far away point, not just on the axis. This suggests that a circular current loop behaves as a magnetic dipole of magnetic moment,

$m = IA$

• Thus, the magnetic dipole moment of any current loop is equal to product of the current and its loop area.