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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

B =    

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 I1 and I2 running in parallel directions. The field from the first wire has a strength of

B1 =

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.
Figure %: The magnetic field on one wire caused by another wire running parallel to it
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 I1 and I2. 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.

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