Two electrons each of charge e and mass m are attached one at each end of a light rigid rod of length 2r.The rod is rotated at constant angular speed about a perpendicular axis passing through its centre. What is the ratio of angular momentum about the axis of rotation to the magnetic dipole moment of the system?
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The electrons are moving in a circle of radius r. Let linear velocity be v and angular velocity be ω.
Angular momentum of each electron about the axis through
the center of rod = L = m v r = m r² ω
For both electrons = L = 2 m r² ω
The rotating electrons create (effectively) a current along the circle. It is like a current carrying conducting loop. The value of current = i = 2e * revolutions/sec.
i = 2 e * ω/(2π) = e ω/π
The magnetic dipole moment due to this current loop = M
M = i * area of loop = π r² i = e ω r²
Ratio L/M = 2 m r² ω/ (e ω r² ) = 2 m/e
Angular momentum of each electron about the axis through
the center of rod = L = m v r = m r² ω
For both electrons = L = 2 m r² ω
The rotating electrons create (effectively) a current along the circle. It is like a current carrying conducting loop. The value of current = i = 2e * revolutions/sec.
i = 2 e * ω/(2π) = e ω/π
The magnetic dipole moment due to this current loop = M
M = i * area of loop = π r² i = e ω r²
Ratio L/M = 2 m r² ω/ (e ω r² ) = 2 m/e
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