Question

If you had two solenoids of identical wire and number of turn per unit length, but different radial size, what would you expect to find if you measured the magnetic field inside at the center?

Answer 1

We'll assume you are talking about a solenoidal electromagnet made up of many turns of conducting wire (say, copper) wound around a cylinder with a length that is much longer than the diameter.

The magnetic field at any point in space can be computed by summing over the magnetic fields produced by each turn of wire in your solenoid. It turns out that for an infinitely long solenoid, with the same number of turns per unit length of the solenoid, the magnetic field is constant in strength everywhere inside. If your solenoid has ends, then you can think of it as an infinitely long solenoid minus the end parts that stretch off to infinity. The magnetic field strength on the axis going right through the solenoid, in the place on the end of the solenoid is then the field of an infinitely long solenoid minus half of it because half is missing, and so the field strength is half as big on the ends (but right in the middle).

The field strength in the middle of a long solenoid is almost exactly that of an infinitely long solenoid, or twice that on the ends.

The field lines really have to go around in loops because they cannot begin or end anywhere (there are no magnetic charges). Field lines penetrate through the coils and the field starts pointing out from the ends of the solenoid and turning around to go back in the other end of the solenoid. We often call this field the "fringe field" of the solenoid.

You can modify the field shape by wrapping your solenoid around an iron core, and if the iron core loops back around to go back in the other end of the solenoid, the fringe field can be reduced. Iron has a large magnetic permeability, and magnetic field lines prefer to stay inside the iron. So the magnetic field strength in this case would almost be the same on the ends of the solenoid as in the center.

If the solenoid is made out of a superconducting sheet, or tightly wrapped superconducting wire, then the field strength will also be the same or nearly so at the ends as in the middle. The reason for this is that a superconductor expels magnetic fields from its bulk, so magnetic field lines cannot stray through the coils and "leak" out the sides.

Magnetic flux is the surface integral of the magnetic field over an area. If the magnetic field has a lower strength, then the magnetic flux will be less as well.