A particle of charge q is revolving in a circle of radius with a const...
Introduction
To find the ratio of the magnitudes of the magnetic moment (μ) and angular momentum (L) of a charged particle revolving in a circle, we start by defining these quantities.
Magnetic Moment (μ)
- The magnetic moment of a current loop is given by the formula: μ = I * A, where I is the current and A is the area.
- For a charged particle of charge q moving with speed v in a circular path of radius r:
- Current (I) = q / T, where T is the period of rotation.
- The period T can be expressed as T = 2πr / v.
- Therefore, I = qv / (2πr).
- The area (A) of the circular path is A = πr².
- Thus, the magnetic moment can be calculated as:
- μ = (qv / (2πr)) * (πr²) = (qvr / 2).
Angular Momentum (L)
- The angular momentum of a particle moving in a circle is given by: L = mvr, where m is the mass of the particle.
Ratio of Magnetic Moment to Angular Momentum
- To find the ratio μ / L:
- μ = (qvr / 2)
- L = mvr
- Therefore, the ratio becomes:
- μ / L = [(qvr / 2) / (mvr)] = (q / (2m)).
Conclusion
- The ratio of the magnitudes of magnetic moment to angular momentum for a charged particle in circular motion is:
- μ / L = q / (2m).
- This indicates that the magnetic moment is directly proportional to the charge and inversely proportional to the mass of the particle.
A particle of charge q is revolving in a circle of radius with a const...
q/2m