Question

One of the methods physicists have used to search for magnetic monopoles is to monitor the current produced in a loop of wire. Draw graphs of current in the loop vs. time for an electrically uncharged magnetic monopole passing through the loop, and for an electrically uncharged magnetic dipole (such as a neutron) passing through the loop with its north end head-first. Don’t worry too much about the details of the exact moment when the particle goes through the plane of the loop; concentrate on the times just before and just after this event. Explain the differences in the two graphs.

Answer 1

CASE 1: electrically uncharged magnetic monopole passing through the loop

A monopole passing through a wire loop of radius R would induce a current due to Faraday’s Law. Assuming that a single magnetic ‘charge’ emits a magnetic field that obeys a Coulomb-like law, that is

we can work out the magnetic flux through the loop as a single monopole falls through it.

If the speed of the monopole is and it falls along the axis of the loop then, assuming the loop has radius and we take as the area of integration the flat circle within the loop, we need to work out to get the flux. Suppose the monopole is a distance from the centre of the loop. Then the distance from the monopole to a point on the disk with radius is so the field strength at that point is

The term isolates the component of that is perpendicular to the disk, which is where is the angle between the axis and a line from the monopole to a point on the disk at radius . We get

so the flux from the monopole is

If we take to be the time when the monopole crosses the plane of the disk and take the surface normal at the disk to point in the direction of the monopole’s velocity, then for and for , and for and for . That is

To go further, we need to modify Maxwell’s equations to include magnetic charge. The relevant one is Faraday’s law which needs an extra term:

where is the magnetic current density. This is the analog to the equation which involves electric current density. Applying Stokes’s theorem, we can integrate the LHS around the loop and the RHS over the disk enclosed:

where is the induced back emf and is the magnetic current flowing through the loop. This emf can be written in terms of the self-inductance of the loop:

What are we to make of considering we have only a single monopole to make up the current? We can write it as a delta function:

That is, there is a current consisting of a single charge across the disk only at time . Since the delta function is the derivative of the step-function , we can integrate Faraday’s law to get

So we get

The term comes out to be the same as the term, so we get in general

One can easily draw the graph for the above equation. For , , while for , . time constant of the curve will be 'b/v'.

CASE 2: electrically uncharged magnetic dipole passing through the loop with its north end head-first

Assume that the loop lies on the x-y plane.You can solve this problem either in the general case (zzcan have any value) or for the case of z>>bz>>b where you can assume the field of dipole is uniform on the surface of the loop. We solve for the general case.

Now we find the magnetic field of the dipole on this plane (z=0).We use cylindrical coordinates: