12.5 Ampère’s law  (Page 22/3)

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 $\overrightarrow{B}$ 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.

[link] 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 ${\displaystyle \oint \overrightarrow{B}}·d\overrightarrow{l}$ 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, $\overrightarrow{B}·d\overrightarrow{l}$ is the product of B and the projection of dl onto the circle passing through $d\overrightarrow{l}.$ If the radius of this particular circle is r , the projection is $rd\theta ,$ and

With $\overrightarrow{B}$ given by [link] ,

For path M , which circulates around the wire, ${\displaystyle {\oint }_{M}d\theta }=2\pi$ and

Path N , on the other hand, circulates through both positive (counterclockwise) and negative (clockwise) $d\theta$ (see [link] ), and since it is closed, ${\displaystyle {\oint }_{N}d\theta }=0.$ Thus for path N ,

The extension of this result to the general case is Ampère’s law.

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 [link] . 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.

Using ampère’s law to calculate the magnetic field due to a wire

Use Ampère’s law to calculate the magnetic field due to a steady current I in an infinitely long, thin, straight wire as shown in [link] .

Strategy

Consider an arbitrary plane perpendicular to the wire, with the current directed out of the page. The possible magnetic field components in this plane, $B_r$ and $B_{\theta },$ are shown at arbitrary points on a circle of radius r centered on the wire. Since the field is cylindrically symmetric, neither $B_r$ nor $B_{\theta }$ varies with the position on this circle. Also from symmetry, the radial lines, if they exist, must be directed either all inward or all outward from the wire. This means, however, that there must be a net magnetic flux across an arbitrary cylinder concentric with the wire. The radial component of the magnetic field must be zero because ${\overrightarrow{B}}_r\cdot d\overrightarrow{l}=0.$ Therefore, we can apply Ampère’s law to the circular path as shown.

Solution

Over this path $\overrightarrow{B}$ is constant and parallel to $d\overrightarrow{l},$ so

${\displaystyle \oint \overrightarrow{B}}·d\overrightarrow{l}=B_{\theta }{\displaystyle \oint dl}=B_{\theta }(2\pi r).$

Thus Ampère’s law reduces to

$B_{\theta }(2\pi r)={\mu }_{0}I.$

Finally, since $B_{\theta }$ is the only component of $\overrightarrow{B},$ we can drop the subscript and write

$B=\frac{{\mu }_{0}I}{2\pi r}.$

This agrees with the Biot-Savart calculation above.

Significance

Ampère’s law works well if you have a path to integrate over which $\overrightarrow{B}·d\overrightarrow{l}$ has results that are easy to simplify. For the infinite wire, this works easily with a path that is circular around the wire so that the magnetic field factors out of the integration. If the path dependence looks complicated, you can always go back to the Biot-Savart law and use that to find the magnetic field.

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