Ampere’s Circuital Law

Ampere’s circuital law is an alternative to Biot-Savart law. This law is a relationship between the integrated magnetic field around a closed loop to the electric current pasing through the loop. Andró-Marie Ampere, a French mathematician, chemist and philosopher in 1826 discovered this relation which is called Ampere’s law or Ampere’s theorem or Ampere’s circuital law.

This law states that the line integral of magnetic field $\overrightarrow{B}$ around any closed path in vacuum/air is equal to ${\mu }_o$ times the net current enclosed by that path.

Mathematically,

$$\oint \overrightarrow{B}.\overrightarrow{dl}={\mu }_o\sum I$$

Here, ${\mu }_o$ is the permeability of free space.

$\overrightarrow{dl}$ is an infinitesimal element (differential) of the closed path. It a vector with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the path.

$\sum I$ represents the net current inside the loop. The closed path is termed as Amperian loop.

Proof of Ampere’s Circuital Law

Consider a long straight conductor carrying current $I$. Due to current $I$ in the conductor, the magnetic field is produced around the conductor in which the magnetic lines of force are concentric circles with centre at the conductor. The magnitude of the magnetic field at $P$ at a distance $r$ from the conductor is given by,

$$B=\frac{{\mu }_o}{2\pi }\frac{I}{r}\text{ \_\_(1)}$$

Consider a small element $\overrightarrow{dl}$ at point $P$ in a circular magnetic field of radius $r$. On this closed loop, the magnitude of $\overrightarrow{B}$ is same everywhere and the direction of $\overrightarrow{B}$ at every point is along the tangent to the circle. Hence, the direction of $\overrightarrow{B}$ and $\overrightarrow{dl}$ is same so the angle between them is zero. Therefore, the line integral of $\overrightarrow{B}$ over this closed loop is given by,

$$\oint \overrightarrow{B}.\overrightarrow{dl}=\oint Bdl\cos 0=\oint Bdl$$

$$=\oint \frac{{\mu }_{o}I}{2\pi r}dl=\frac{{\mu }_{o}I}{2\pi r}\oint dl=\frac{{\mu }_{o}I}{2\pi r}\times 2\pi r$$

$$={\mu }_{o}I$$

$$\therefore \oint \overrightarrow{B}.\overrightarrow{dl}={\mu }_{o}I$$

Here, $\oint dl=2\pi r=$circumference of the circle.

This proves Ampere’s circuital law.

This relation shows that the magnetic field is independent of $r$ i.e. size and shape of the closed curve enclosing a current or sum of currents. If current $I_2$ is passing in opposite to $I_1$, the currents enclosed by the loop is $(I_1-I_2)$. Then, Ampere’s law is given by,

$$\oint \overrightarrow{B}.\overrightarrow{dl}={\mu }_o(I_1-I_2)$$

In order to use this law, it is necessary to choose a closed path for which it is possible to determine the line integral of $\overrightarrow{B}$. For this reason, this law has a limited use. This law is not a universal law. If it is not convenient to use this law, we use Biot-Savart law to determine $\overrightarrow{B}$.

This law is true only for steady currents and is useful only for calculating the magnetic field of current configuration having a high degree of symmetry.

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Next: Applications of Ampere’s Circuital Law