0.10 Ampere's law

Ampere law supplements Biot-Savart law in providing relation between current and magnetic field. Biot-Savart law provides expression of magnetic field for a small current element. If we need to find magnetic field due to any extended conductor carrying current, then we are required to use techniques like integration and superposition principle. Ampere law is another law that relates magnetic field and current that produces it. This law provides some elegant and simple derivation of magnetic field where derivation using Biot-Savart law would be a difficult proposition. This advantage of Ampere law lies with the geometric symmetry, which is also its disadvantage. If the conductor or circuit lacks symmetry, then integration involving Ampere’s law is difficult.

Ampere law as modified by Maxwell for displacement current is one of four electromagnetic equations.

Basis of ampere law

In order to understand the basis of Ampere law, we investigate here the magnetic field produced by a straight conductor carrying current. The expression of magnetic field due to long straight (infinite) conductor carrying current as obtained by applying Biot-Savart law is :

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

where R is the perpendicular distance between straight conductor and point of observation. Rearranging, we have :

$$\Rightarrow 2\pi RB={\mu }_{0}I$$

If we carefully examine the left hand expression, then we find that it is an integral of the scalar product of magnetic field and length element about the perimeter of a circle drawn with center on the straight conductor and point of observation lying on it.

$$\int \mathbf{B}.đ\mathbf{l}={\mu }_{0}I$$

Now, we evaluate the left hand integral to see whether our observation is correct or not? For the imaginary circular path, the direction of length element and magnetic field are tangential to the circle. The angle between two vector quantities is zero. Hence, left hand side integral is :

$$\int \mathbf{B}.đ\mathbf{l}=\int Bđl\cos 0°=\int Bđl$$

Integration along circular path

Since magnitude of magnetic field due to current in straight wire are same at all points on the circular path - being at equal distance from the center, we take magnetic field out of the integral,

$$\Rightarrow \int \mathbf{B}.đ\mathbf{l}=B\int đl=2\pi RB$$

Substituting in the equation of line integral of magnetic field as formulated earlier, we have the same expression of magnetic field for long straight conductor as obtained by applying Biot-Savart law :

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

It is clear here that the left hand side integration should be carried out over a “closed” path. This closed path is termed as “closed imaginary line” or “Ampere loop”. Hence, we write the equation as :

$$\oint \mathbf{B}.đ\mathbf{l}={\mu }_{0}I$$

Note the circle in the middle of integration sign which indicates a closed path of integration. This formulation is evidently an alternative to Biot-Savart law in the instant case. Now the question is whether this relation is valid for any “closed imaginary line”? The answer is yes. Though the above equation involving closed line integral is valid for any closed imaginary path, but only few of these closed paths allow us to use the equation for determining magnetic field. For instance, if we consider a square path around the straight wire, then we face the problem that points on the path are not equidistant from the wire and as such magnetic field is not same as in the case of a circular path. It is also evident that we need to choose a loop which passes through the point of observation. After all, we are interested to know magnetic field due to currents at a particular point in a region. See Ampere's law(exercise) : Problem 1 which illustrates this aspect of application of Ampere's law.