Magnetic Poles, Forces, and the Biot-Savart Law

The purpose of this note is to show how to use the concept of the magnetic pole. It is useful for deriving Biot-Savart law and a few other things that are probably not discussed in your text.

The point magnetic pole, analogous to the point charge, does not really exist. Don't let that bother you; it is a useful foil. The constant m_o/4p is analogous to 1/4pe_o. Thus if we define pole strength q_m for the ideal magnetic point pole, its magnetic field is (m_o/4p)q_m/r² and if it is in an external field B, the force on it is Bq_m (analogous to F = Eq).

To find the net force on a magnet (two poles) in the neighborhood of another magnet, one would add four vectors: the force on each pole due to each pole of the other magnet.

Another useful aspect to the idea of a point pole is that it provides us with an easy derivation of the Biot-Savart law. Consider a charge q with velocity v. We wish to find the magnetic field that it produces as a function of position. So imagine a magnetic pole at some point not collinear with v. The force on that pole will be equal and opposite to the force that the pole exerts on q. Of course the force on q is qvxB, and B is (m_o/4p)q_m/r² in the r direction. The field at the location of q_m due to the moving q is F/q_m. Do the grubby details about the directions of these vectors and show that B = (m_o/4p)qvxr/r³, where r is from q to the point where our hypothetical pole is located.

Now consider the field of a circuit element idl. It consists of charge dq moving distance dl in time dt. Show that (dq)v = idl and hence dB = (m_o/4p)idlxr/r³.

We have used two fundamental principles: the inverse square law for the field from a pole, and the force on a moving charge. The equation F=qvxB defines the unit for B, F= q_m B defines the unit for q_m.

Incidentally, physicists have long speculated about the possibility of the existence of elementary particles possessing a single pole (magnetic monopoles). If you find one, you will get the Nobel Prize.

The problem with the idea of point poles is that real magnets have poles spread out over an area. As in the case of smeared-out charges, we can calculate exact forces, etc. only in some special cases. For example, we can find the force of attraction of parallel plates with equal and opposite charges if the plates are close together compared to length and width. Similarly, we can find the force of attraction between parallel opposite poles. For the charged plates, the force is the charge of one times the field due to the other. So F = q(½E). Recall E = s/e_o, so q = e_oEA. Then the force becomes F = ½e_oE²A. Similarly for flat, close parallel magnetic opposite poles, F = B²A/2m_o. (e_o is analogous to 1/m_o.)

Let us check this by an energy approach. If you move the poles farther apart by an amount dx, the work done is Fdx. This increases the energy in the field by (B²/2m_o)(Adx) (i.e., energy/volume times volume). Now equate these and cancel the dx and we get the same F.

You may get the impression that for any force on a charge, the corresponding force on a pole may be found by replacing E with B and e_o with 1/m_o. True if it is the Coulomb force. But consider a charge moving in a magnetic field: F = qvxB. The force on a magnetic pole moving in an electric field is

F = -m_oe_oq_mvxE.

Try to derive this, using the equal and opposite forces similar to what was done in the derivation of the Biot-Savart law.

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