Lesson: Chapter - 15

The Magnetic Field Due to a Current

So far we have discussed the effect a magnetic field has on a moving charge, but we have not discussed the reverse: the fact that a moving charge, or current, can generate a magnetic field. There’s no time like the present, so let’s get to it.

The magnetic field created by a single moving charge is actually quite complicated, and is not covered by Physics. However, the magnetic field created by a long straight wire carrying a current, I, is relatively simple, and is fair game for Physics. The magnetic field strength is given by:

B = µ₀I/2pr

Video Lesson

The constant µ₀ is called the permeability of free space, and in a vacuum it has a value of about 4p × 10⁷ N/A².

For Physics, it’s not important to memorize this equation exactly. It’s more important to note that the strength of the magnetic field is proportional to the strength of the current and is weaker the farther it is from the wire.

The direction of the magnetic field lines are determined by an alternate version of the right-hand rule: if you held the wire with your thumb pointing in the direction of the current, the magnetic field would make a circular path around the wire, in the direction that your fingers curl.

Example

Two parallel long straight wires carrying a current I stand a distance r apart. What force does one wire exert on the other?

Consider the magnetic field created by the bottom wire as it affects the top wire. According to the right-hand rule, the magnetic field will point out of the page, and will have a strength of B = (µ₀ I)/(2pr).

The force exerted by the bottom wire on the top wire is F = IlB. If we substitute in for B the equation we derived above, we find the force per unit length is:

Using the right-hand rule once more, we find that the force pulls the top wire down toward the bottom wire.

We can apply the same equations to find that the top wire pulls the bottom wire up. In other words, the two wires generate magnetic fields that pull one another toward each other. Interestingly, the fact that each wire exerts an opposite force on the other is further evidence of Newton’s Third Law.

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