^{*}

^{*}

We consider the effect of a magnetic field on the motion of an atomic electron in its orbit. The usual treatment deals with the change in magnetic dipole moment assuming the electron's speed changes but the radius of its orbit remains unchanged. We derive the change in the magnetic dipole moment allowing both the speed and the radius to change. The cases of fixed radius on one hand and of fixed speed on the other are treated as special cases of our general case.

Typically, classical electromagnetism predicts a change in the magnetic dipole moment of an orbital electron when an external magnetic field is set up normal to the plane of the electron’s orbit [1, 2]. It is usually assumed that the speed of the electron changes but the radius remains unchanged during the time through which the magnetic field is changing [

Consider an atomic electron that rotates counterclockwise with a speed in a circular orbit, in the x-y plane, of radius. In practice the period of the electron’s motion is so small that one may assume it constitutes a steady current:

and thus its orbital dipole moment is

where is the charge of the electron. The electric force exerted on the electron by the nucleus (of charge) is balanced by the centripetal force,

where is the mass of the electron. Now, introduce a magnetic field in the z direction, of strength, which, initially zero, increases to. The form of is irrelevant provided that it varies sufficiently slowly, which means on a time scale long compared to the electron’s orbital period. The magnetic force exerted on the electron points toward the center and of magnitude, so the centripetal force must be modified to accommodate for magnetic force. Thus

where we allow both the speed and the radius of the orbit to change from to and to respectively. Letting and, Equation (4) reads

which, with the help of Equation (3), can be written as

Using binomial expansion to first order in and and neglecting terms containing the product, Equation (6) yields

which, with a further use of binomial expansion for the left hand side term, gives

Our result in Equation (8) is the general case which allows changes in the speed and the radius of the orbit. Before we proceed further, we need to consider the two special cases in the following section.

Our aim here is utilize our result of Equation (8) for two special cases: the first deals with fixed orbit () but changes and the second deals with fixed speed () but changes.

For, Equation (8) immediately gives

which corresponds to a change in the magnetic dipole moment predicted by Equation (2), namely

The above result is the same as that derived by Griffiths in ref. [

One may also derive and thus by considering the angular impulse exerted by the induced electric field which is created during the change of the magnetic field. Faraday’s law of induction gives an induced electric field given by

and thus the torque exerted on the electron is

which immediately gives the change in angular momentum, that is

Therefore, upon using, one gets the correct change in the magnetic moment given in Equation (10). It is interesting to calculate the work, W done by the induced electric field:

where we used for the average speed.

Using the expression for in Equation (2) and that of in Equation (10), the work done takes the form

Alternatively, one may calculate the change in the kinetic energy of the electron,

which upon the substitution of from Equation (9), from Equation (2) and from Equation (10), the change in the kinetic energy takes the form

The above result for is equal to the work done given in Equation (17) and thus the work-energy theorem is verified.

For, Equation (8) gives

and in this case the change in the magnetic moment is given by

which upon using Equation (19) becomes

Obviously, our result shows that the magnitude of the change of the magnetic moment for fixed speed is twice its magnitude for the fixed radius case. The result in Equation (21) is what Griffiths claimed in reference [

Going back to the general case in which both the speed and the radius are changing, one may rewrite Equation (8) in the form

In the present case, the variation in the magnetic moment is

which upon the use of Equation (23) it becomes

Note that the last term on the right hand-side looks like the same as for the fixed radius case. Therefore, using Equation (9), we get

The above result is exactly the same as that for the fixed radius case, and thus it corresponds to the same change in the angular momentum which is given in Equation (14).

In this work, the effect of an external magnetic field on an atomic electron has been examined. The usual treatment of this problem assumes (without justification) a fixed radius for the orbit but allow the electron’s speed to change. In the present paper we considered the general case in which both the speed and the radius are allowed to change. We used binomial expansion to first order in the change of speed and in the change of the radius. The two special cases for fixed orbit and for fixed speed were deduced from our general case and in each of these two cases the change in the magnetic moment and the change in the angular momentum have been derived. Interestingly, our result for the general case yields a change in the magnetic moment which is the same as that for the fixed radius case.