A Derivation of the Electromagnetic Mass and Self-Torque of an Infinitely Long Solenoid

G.R.Dixon, 4/30/2005

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 of s. 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

, (1)

where I_{enc} signifies the current through a rectangle of 1 meter height and enclosing a section of the solenoid wall.

I_{enc} can be expressed in other terms as

. (2)

Substituting in Eq. 1:

. (3)

The energy density in the magnetic field is

. (4)

Thus the magnetic field energy in a unit length of the solenoid is

. (5)

As in the case of a translating spherical shell of charge, E_{B} is presumably the (rotational) kinetic energy of a unit length of the solenoid:

, (6)

where the units of z_{ElecMag} are kg/meter. Solving for z_{ElecMag}:

. (7)

We can check Eq. 7 by noting that the angular momentum per unit length points in the positive y-direction and has the magnitude:

. (8)

Let us suppose that, at time t=0, an w-reducing angular acceleration, a, occurs such that

. (9)

By Newton, the applied torque per unit length must be

. (10)

The (rotational) impulse delivered in one second is

, (11)

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:

. (12)

But from Maxwell such a nonzero dB/dt induces a tangential electric field such that

. (13)

That is,

. (14)

The charge per unit length is

. (15)

Thus there is an a-induced self-torque per unit length of magnitude

. (16)

The externally applied torque per unit length is

. (17)

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 –d__B__/dt (i.e. in the positive y-direction).