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#Maxwell s Equations: Ampere s Law

[Equation 2]

Equation [2] can be explained: Suppose you have a conductor (wire) carrying a current, I. Then this current produces a Magnetic Field which circles the wire.

The left side of Equation [2] means: If you take any imaginary path that encircles the wire, and you add up the Magnetic Field at each point along that path, then it will numerically equal the amount of current that is encircled by this path (which is why we write for encircled or enclosed current).

Let's do an example for fun. Suppose we have a long wire carrying a constant electric current, I [Amps]. What is the magnetic field around the wire, for any distance r [meters] from the wire?

Let's look at the diagram in Figure 1. We have a long wire carrying a current of I Amps. We want to know what the Magnetic Field is at a distance r from the wire. So we draw an imaginary path around the wire, which is the dotted blue line on the right in Figure 1:

Figure 1. Calculating the Magnetic Field Due to the Current Via Ampere's Law.

Ampere's Law [Equation 2] states that if we add up (integrate) the Magnetic Field along this blue path, then numerically this should be equal to the enclosed current I.

Now, due to symmetry, the magnetic field will be uniform (not varying) at a distance r from the wire. The path length of the blue path in Figure 1 is equal to the circumference of a circle of radius r. .

If we are adding up a constant value for the magnetic field (we'll call it H ), then the left side of Equation [2] becomes simple:

[Equation 3]

Hence, we have figured out what the magnitude of the H field is. And since r was arbitrary, we know what the H-field is everywhere. Equation [3] states that the Magnetic Field decreases in magnitude as you move farther from the wire (due to the 1/r term).

So we've used Ampere's Law (Equation [2]) to find the magnitude of the Magnetic Field around a wire. However, the H field is a Vector Field. which means at every location is has both a magnitude and a direction. The direction of the H-field is everywhere tangential to the imaginary loops, as shown in Figure 2. The right hand rule determines the sense of direction of the magnetic field:

Manipulating the Math for Ampere's Law

We are going to do the same trick with Stoke's Theorem that we did when looking at Faraday's Law. We can rewrite Ampere's Law in Equation [2]: