Chapter 12. Sources of Magnetic Fields
12.5 Ampère’s Law
Learning Objectives
By the end of this section, you will be able to:
- Explain how Ampère’s law relates the magnetic field produced by a current to the value of the current
- Calculate the magnetic field from a long straight wire, either thin or thick, by Ampère’s law
A fundamental property of a static magnetic field is that, unlike an electrostatic field, it is not conservative. A conservative field is one that does the same amount of work on a particle moving between two different points regardless of the path chosen. Magnetic fields do not have such a property. Instead, there is a relationship between the magnetic field and its source, electric current. It is expressed in terms of the line integral of [latex]\stackrel{\to }{\textbf{B}}[/latex] and is known as Ampère’s law. This law can also be derived directly from the Biot-Savart law. We now consider that derivation for the special case of an infinite, straight wire.
Figure 12.14 shows an arbitrary plane perpendicular to an infinite, straight wire whose current I is directed out of the page. The magnetic field lines are circles directed counterclockwise and centered on the wire. To begin, let’s consider [latex]\oint \stackrel{\to }{\textbf{B}}·d\stackrel{\to }{\textbf{l}}[/latex] over the closed paths M and N. Notice that one path (M) encloses the wire, whereas the other (N) does not. Since the field lines are circular, [latex]\stackrel{\to }{\textbf{B}}·d\stackrel{\to }{\textbf{l}}[/latex] is the product of B and the projection of dl onto the circle passing through [latex]d\stackrel{\to }{\textbf{l}}.[/latex] If the radius of this particular circle is r, the projection is [latex]rd\theta ,[/latex] and
With [latex]\stackrel{\to }{\textbf{B}}[/latex] given by Equation 12.9,
For path M, which circulates around the wire, [latex]{\oint }_{M}d\theta =2\pi[/latex] and
Path N, on the other hand, circulates through both positive (counterclockwise) and negative (clockwise) [latex]d\theta[/latex] (see Figure 12.14), and since it is closed, [latex]{\oint }_{N}d\theta =0.[/latex] Thus for path N,
The extension of this result to the general case is Ampère’s law.
Ampère’s law
Over an arbitrary closed path,
where I is the total current passing through any open surface S whose perimeter is the path of integration. Only currents inside the path of integration need be considered.
To determine whether a specific current I is positive or negative, curl the fingers of your right hand in the direction of the path of integration, as shown in Figure 12.14. If I passes through S in the same direction as your extended thumb, I is positive; if I passes through S in the direction opposite to your extended thumb, it is negative.
Problem-Solving Strategy: Ampère’s Law
To calculate the magnetic field created from current in wire(s), use the following steps:
- Identify the symmetry of the current in the wire(s). If there is no symmetry, use the Biot-Savart law to dete