Permanent magnets

Permanent magnets are the most familiar sources of magnetic fields. A compass needle is a permanent magnet, which itself reacts to the permanent magnet in the earth's axis. Unfortunately, the fields of permanent magnets are very hard to calculate, and require an understanding of complex ferromagnetic phenomena, belonging as much to atomic theory as to electromagnetism. Here we will simply give a qualitative description of the magnetic fields of permanent magnets.

In essence, a permanent magnet is a piece of metal with a "North Pole" and a "South Pole". Any magnetized piece of metal has both poles; no magnet can exist with only one pole. Since magnetic charge does not exist, there is no isolated concentration of magnetic charge in an object. So why not simply take a magnet and split it in half, thus separating the north and south ends? Well, when we try it, two smaller, identical magnets are produced, shown below. Again, the north or south end of a magnet cannot be isolated.

Even though we cannot describe quantitatively the field of a permanent magnet, we can show its shape:

The field lines always point away from the north end, and toward the south end, in a shape similar to the electric field between two oppositely charged particles. As we will see, this field is quite similar to the field created by a coil of wire with a current running through it (see solenoid). Permanent magnets are often used to create magnetic fields; these magnets are usually oriented in a manner that causes a uniform field, so we do not have to concern ourselves too much with their field shape.

The Magnetic Field of a Straight Wire

Like magnets, current-carrying wires also create magnetic fields. Wires of and any and all shapes create a magnetic field, but straight wires are the easiest to work with. After going through some calculus we will tackle more complex situations, but for now we look at the most simple case: the straight wire.

Shape of the Field

As we know, the magnetic field must always be perpendicular to the direction of the current; in terms of a field around a wire, this means that the field lines must follow a circular path about the wire, as shown below.

Given that the field lines travel in a circle around the wire, as shown, how do we decide which way the field lines point? We use our hands again, relying on the second right hand rule. Take your right hand, stick your thumb up like a hitchhiker, curling your fingers around. If you point your thumb in the direction of the current, your fingers will curl around in the direction of the field lines. Try it with the figure above--in many ways this right hand rule is simpler than our first right hand rule.

Magnitude of the Field

At a point a distance r away from a wire carrying a current I , the magnetic field has been experimentally measured to have a value of:

straightwireeq

As we explained above, this field points perpendicular to the current, in a circle around the wire. This equation indicates that the strength of the magnetic field decreases as one gets farther away from the wire; it varies with 1/r . In addition, a stronger current causes a greater magnetic field, as expected.

Given this equation, we can calculate the phenomenon of attraction and repulsion Oersted saw in the interactions between two wires. Consider two wires, separated by a distance r , with currents I ₁ and I ₂ running in parallel directions. The field from the first wire has a strength of

near the second wire. The direction of this field, according to our second right hand rule, points perpendicular to the plane of the two wires, as shown below. Since we now have the strength of the field on the wire, and we know the force on a wire from a given magnetic field, we can calculate the force on the second wire, per unit length:

The direction of this force, according to the first right hand rule, is towards the other wire. Notice that the equation is symmetric in I ₁ and I ₂ . Indeed, the same equation governs the force on the first wire from the second, as we would expect from Newton's Third Law. We have thus derived the attractive force between wires, one of the first indications of electromagnetism.

Having dealt with the simplest sources of magnetic fields, we must now tackle the more difficult ones, such as odd-shaped wires, and rings, and coils. This endeavor will require some calculus, which we will establish in the next section.