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Question

An electron (e,m) is revolving in a circular orbit about the nucleus of an atom. Find a relationship between the orbital magnetic dipole moment μ and the angular momentum L of the electron about the centre of its orbit.

A
μ=emL
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B
μ=emL
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C
μ=e2mL
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D
μ=e2mL
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Solution

The correct option is C μ=e2mL
Let v be the speed of the electron and r be the radius of its orbit.

The orbiting electron creates an electric current opposite to the velocity.

Thus, current I is charge e times the frequency f of the orbital motion.

I=ef=eT=ev2πr



The magnitude of magnetic moment,

μ=IA=ev2πrπr2=evr2 ....(1)

The direction of the magnetic dipole moment is perpendicular to the loop.

The orbital angular momentum is given by

Lorbital=r×p

|Lorbital|=rpsin90=mvr ...(2)

The direction of the orbital angular momentum is determined from the vector product right-hand rule.

The direction of angular momentum and magnetic dipole moment is opposite to each other. Comparing eqns. (2) and (1), we obtain

μ=e2mL

Hence, option (c) is the correct answer.

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