4.7 AMPERE’S CIRCUITAL LAW

4.14

There is an alternative and appealing way in which the Biot-Savart law may be expressed. Ampere’s circuital law considers an open surface with a boundary (Fig. 4.14). The surface has current passing through it. We consider the boundary to be made up of a number of small line elements. Consider one such element of length $d l$ . We take the value of the tangential component of the magnetic field, $B_{t}$ , at this element and multiply it by the length of that element $d l$ [Note: $B_{t} d l = B . d l$ ]. All such products are added together. We consider the limit as the lengths of elements get smaller and their number gets larger. The sum then tends to an integral. Ampere’s law states that this integral is equal to $μ_{0}$ times the total current passing through the surface, i.e.,

$\oint B \cdot d l = μ_{0} I$ (4.17(a))

where $I$ is the total current through the surface. The integral is taken over the closed loop coinciding with the boundary $C$ of the surface. The relation above involves a sign-convention, given by the right-hand rule. Let the fingers of the right-hand be curled in the sense the boundary is traversed in the loop integral $\oint B \cdot d l$ . Then the direction of the thumb gives the sense in which the current $I$ is regarded as positive.

For several applications, a much simplified version of Eq. [4.17(a)] proves sufficient. We shall assume that, in such cases, it is possible to choose the loop (called an amperian loop) such that at each point of the loop, either

$B$ is tangential to the loop and is a non-zero constant B, or
$B$ is normal to the loop, or
$B$ vanishes.

Now, let $L$ be the length (part) of the loop for which $B$ is tangential. Let $I_{e}$ be the current enclosed by the loop. Then, Eq. (4.17) reduces to,

$B L = μ_{0} I_{e}$ (4.17(b))

When there is a system with a symmetry such as for a straight infinite current-carrying wire in Fig. 4.15, the Ampere’s law enables an easy evaluation of the magnetic field, much the same way Gauss’ law helps in determination of the electric field. This is exhibited in the Example 4.9 below. The boundary of the loop chosen is a circle and magnetic field is tangential to the circumference of the circle. The law gives, for the left hand side of Eq. [4.17 (b)], $B . 2 π r$ . We find that the magnetic field at a distance $r$ outside the wire is tangential and given by

$\begin{array}{l} B \times 2 π r = μ_{0} I \\ B = μ_{0} I / (2 π r) \end{array}$ (4.18)

The above result for the infinite wire is interesting from several points of view.

It implies that the field at every point on a circle of radius $r$ , (with the wire along the axis), is same in magnitude. In other words, the magnetic field possesses what is called a cylindrical symmetry. The field that normally can depend on three coordinates depends only on one: $r$ . Whenever there is symmetry, the solutions simplify.
The field direction at any point on this circle is tangential to it. Thus, the lines of constant magnitude of magnetic field form concentric circles. Notice now, in Fig. 4.1(c), the iron filings form concentric circles. These lines called magnetic field lines form closed loops. This is unlike the electrostatic field lines which originate from positive charges and end at negative charges. The expression for the magnetic field of a straight wire provides a theoretical justification to Oersted’s experiments.
Another interesting point to note is that even though the wire is infinite, the field due to it at a non-zero distance is not infinite. It tends to blow up only when we come very close to the wire. The field is directly proportional to the current and inversely proportional to the distance from the (infinitely long) current source.
There exists a simple rule to determine the direction of the magnetic field due to a long wire. This rule, called the right-hand rule*, is:
Grasp the wire in your right hand with your extended thumb pointing in the direction of the current. Your fingers will curl around in the direction of the magnetic field.

*Note that there are two distinct right-hand rules: One which gives the direction of $B$ on the axis of current-loop and the other which gives direction of $B$ for a straight conducting wire. Fingers and thumb play different roles in the two.

Ampere’s circuital law is not new in content from Biot-Savart law. Both relate the magnetic field and the current, and both express the same physical consequences of a steady electrical current. Ampere’s law is to Biot-Savart law, what Gauss’s law is to Coulomb’s law. Both, Ampere’s and Gauss’s law relate a physical quantity on the periphery or boundary (magnetic or electric field) to another physical quantity, namely, the source, in the interior (current or charge). We also note that Ampere’s circuital law holds for steady currents which do not fluctuate with time. The following example will help us understand what is meant by the term enclosed current.

Example 4.8

Figure 4.15 shows a long straight wire of a circular cross-section (radius $a$ ) carrying steady current $I$ . The current $I$ is uniformly distributed across this cross-section. Calculate the magnetic field in the region $r < a$ and $r > a$ .

4.15

VIEW SOLUTION

(a) Consider the case $r > a$ . The Amperian loop, labelled 2, is a circle concentric with the cross-section. For this loop, $L = 2 π r$ $I_{e} =$ Current enclosed by the loop $= l$
The result is the familiar expression for a long straight wire

$\begin{array}{l} B (2 π r) = μ_{0} I \\ B = \frac{μ_{0} I}{2 π r} \end{array}$ (4.19(a))

(b) Consider the case $r < a$ . The Amperian loop is a circle labelled 1. For this loop, taking the radius of the circle to be $r$ , $L = 2 π r$

Now the current enclosed $I_{e}$ is not $I$ , but is less than this value. Since the current distribution is uniform, the current enclosed is, $I_{e} = I (\frac{π r^{2}}{π a^{2}}) = \frac{I r^{2}}{a^{2}}$

Using Ampere’s law, $B (2 π r) = μ_{0} \frac{I r^{2}}{a^{2}}$

$\begin{array}{l} B = (\frac{μ_{0} I}{2 π a^{2}}) r \\ B \propto r (r < a) \end{array}$ (4.19(b))

4.16

Figure (4.16) shows a plot of the magnitude of $B$ with distance $r$ from the centre of the wire. The direction of the field is tangential to the respective circular loop (1 or 2) and given by the right-hand rule described earlier in this section.

This example possesses the required symmetry so that Ampere’s law can be applied readily.

It should be noted that while Ampere’s circuital law holds for any loop, it may not always facilitate an evaluation of the magnetic field in every case. For example, for the case of the circular loop discussed in Section 4.6, it cannot be applied to extract the simple expression $B = μ_{0} I / 2 R$ [Eq. (4.16)] for the field at the centre of the loop. However, there exists a large number of situations of high symmetry where the law can be conveniently applied. We shall use it in the next section to calculate the magnetic field produced by two commonly used and very useful magnetic systems: the solenoid and the toroid.