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Login NowThe magnetic dipole moment is defined as the product of current the loop and the area of loop.

\( \tau \) = IA

The expression for torque can now be written as

\( \tau \) = μβ sinθ ………….. (3)

Equation (2) and (3) give the magnitude of the torque but they do not specify the direction of τ. The direction of the torque can be obtained by expressing equation (3) in vector form.

\( \tau \) = \( \vec{m} \times \vec{B} \) ………………. (4)

Because a magnetic dipole experiences a torque when placed in an external field work must be done to change it’s orientation. This work done is also be referred as energy of dipole.

U = \(\int_{\Theta _{i}}^{\Theta _{f}}\)\( \tau \)dθ

= \(\int_{\Theta _{i}}^{\Theta _{f}}\)μβ sinθ dθ

= \(\int_{90° }^{\Theta }\)μβ sinθ dθ

= -μβ cosθ …………….. (5)

This can be expressed in dot product as

U = – \( \vec{m} . \vec{B} \) ……………. (6)

From equation (5) we conclude that,

U_{max} = μβ, when θ = π. that is when μ and β are anti-aligned

U_{min} = -μβ, when θ = 0, that is when μ and β are aligned.

Electrons revolving around atomic nuclei, electrons spinning on their axes, and rotating positively charged atomic nuclei all are magnetic dipoles. The sum of these effects may not be a magnetic dipole. If they do not fully cancel, the atom is a permanent magnetic dipoles, as iron atoms. The same alignment to form ferromagnetic domain also constitute a magnetic dipole.

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