A Derivation of the Electromagnetic Mass and Self-Torque of an Infinitely Long Solenoid
In this article the electromagnetic mass per unit length of an infinitely long, rotating cylinder of charge is analytically derived. Also demonstrated is the self-torque (or reaction torque) experienced by the solenoid in its own, acceleration-induced electric field when its angular speed is decreased.
Mks units are used. The cylinder has a radius R and a positive, surface charge density ofs. It is concentric to and spins around the y-axis. The constant angular rate is w, and w points in the positive y-direction. In brief, the cylinder constitutes an infinitely long solenoid.
The B field inside such a solenoid is single-valued and points in the positive y-direction. (Outside, B=0 everywhere.) Its magnitude is
where Ienc signifies the current through a rectangle of 1 meter height and enclosing a section of the solenoid wall.
Ienc can be expressed in other terms as
Substituting in Eq. 1:
The energy density in the magnetic field is
Thus the magnetic field energy in a unit length of the solenoid is
As in the case of a translating spherical shell of charge,EB is presumably the (rotational) kinetic energy of a unit length of the solenoid:
where the units of zElecMag are kg/meter. Solving for zElecMag:
We can check Eq. 7 by noting that the angular momentum per unit length points in the positive y-direction and has the magnitude:
Let us suppose that, at time t=0, an w-reducing angular acceleration, a, occurs such that
By Newton, the applied torque per unit length must be
The (rotational) impulse delivered in one second is
which is (see Eqs. 8 and 9) the initial angular momentum.
Let us conclude by demonstrating that the reaction torque per unit length is based on electric forces experienced by the solenoid in its own, a-induced electric field. From Eqs. 1 and 2, a constant angular deceleration of a results in a constant rate of decrease in B:
But from Maxwell such a nonzero dB/dt induces a tangential electric field such that
The charge per unit length is
Thus there is an a-induced self-torque per unit length of magnitude
The externally applied torque per unit length is
Thus the external torque and the reaction torques have identical magnitudes. But the external torque points in the negative y-direction, whereas the a-induced torque points in the same direction as –dB/dt (i.e. in the positive y-direction).