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The Gyro magnetic Ratio

Consider a particle of charge Q and mass M describing a circle of radius a with angular velocity ω.

The equivalent current is        i = Q x frequency = Qω /2π                    ... (1)

The current generates a magnetic dipole moment
        m = iπa2 = Qωa2 /2                                ... (2)

Let L be the angular momentum of the same particle in its orbit. The magnitude of L is Mωa2 and its direction is given by the right and screw rule. Mass M is always positive and therefore the directions of L and ω are identical. Vectors m and L are parallel when Q is positive.
For a positively charged particle,
            M = Qa2 ω/2                                ... (3)

Gyro magnetic ratio g = m/L 1/2Qωa2 /Mωa2 = Q/2M                            ... (4)

For a negatively charged particle,
            m = -Qa2ω/2                                ... (5)
            g = m/L = Q/2M                                ... (6)
The negative sign indicates that vectors m and L have opposite direction.

The Gyro magnetic Ratio

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