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Due to a Current Flowing in a Ring Revolving About a Distant Point |
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Magnetic fields are generated by the motion of electrical charges. This is marvelously described by Maxwell's equations. The time rate of change of charge density generates a current and the current generates a time rate of change in the magnetic field intensity.
In 1826 the French scientist André-Marie Ampére established that under steady state conditions the magnetic field intensity H is related to the current density J by the equation
where c is a parameter that turned out to be the speed of light.
It is to be emphasized that this is under steady state conditions.
The motion of the charges may be complex. The charge on a particle may be rotating with the particle's rotation and the particle as a unit may be rotating with respect to another center of rotation.
Magnetic moments are generated by spinning point charges or charges flowing in cirular paths. Now consider charges flowing in a ring of radius r and the ring rotatating about a distant point in an orbit of radius R from the center of the ring , where R>r. Let ω and Ω be the spin rates of rotation of the ring with respect to its axis and with respect to the distant point, respectively.
A particle of charge on the ring has a tangential velocity of ±ωr due to the spinning of the ring and ±ΩR due to the rotation about the distant point. The signs of the velocities are determined by the directions of the rotations. Presume for now that that both directions of rotation are counterclockwise.
Let Q be the charge on the ring. The charge density σ is then Q/(2πr)
The velocity of a particle of charge σdθ on the outer edge of the ring is (Ω(R+r) + ωr) and on the inner edge of the ring it is (Ω(R−r) − ωr).
The magnetic moment of a circular distribution of charge q due to its rotation is proportional to ωr²q, where ω is the rotation rate in radians per second, q is its total charge and r is the radius of the charges' orbits. The density per unit length of the ring of the magnetic moment generated is
Therefore the increment in magnetic moment that comes from an increment rdθ containing a charge σrdθ is vrdθ/2π. The sum of the particle velocities at θ and (θ+π) is
The integration of the product of these terms with charge density from θ = 0 to θ = π is the same as the integration around the ring from θ = 0 to θ = 2π which produces the result
which is the same as if the rotation of the ring were ignored.
Since a spherical shell may be a considered the asymptotic limit of a stack of cylindrical rings the above results apply for a spherical shell, and by extension to a spherical ball.
The magnetic moment varies with distant from the center of the revolution of a charge. For a revolving charge the average magnetic moment will be at a maximum on a circular ridge along the orbit of the revolving charge.
For a system consisting of a rotating charged particle and the particle revolving about a point outside of the particle the magnetic moment with respect to the point is independent of the rotation of the charged particle. The overall magment moment is the sum of the magnetic moments at the two centers of rotation. .
The velocity of the outer edge of the ring is (Ω(R+r) + ωr) and on the inner edge of the ring it is (Ω(R&minus:r) − ωr). The sum of these two is 2ΩR. A counterbalancing effect on average magnetic moment at the center of the ring comes when the ring is diametrically opposed position on the orbit of the ring. The velocity there is −2ΩR. This added to the velocities of the edges of the ring gives a result of zero. That means the effect of the revolving ring charge on the magnetic moment at the center of the ring is zero.
Therefore the magnetic moment of the system is the sum of the magnetic moments at the two centers computed separately.
The average magnetic moment of a system is the sum of the magnetic moments at the source centers computed separately.