Subject :- Applied Physics 2.

Topic :- Laser.

Difficulty :- Medium.

Ampere’s Circuital law states that “the line integral of magnetic field intensity H around a closed path is exactly equal to the direct current enclosed by that path.” The mathematical representation of Ampere’s law is

$$\vec{H}.{dl}=I \tag{1}$$

The law is very useful to determine $\vec{H}$ when the current distribution is symmetrical.

Since, $\vec{B}=\mu\vec{H}$

Equation (1) can be written as $$\oint\vec{B}.{dl}=\mu_0I$$

This is called the integral form of Ampere’s circuital law

Replacing $ I=\int_S\vec{J}.\vec{ds}$ where $\vec{J}$ is current density and $S$ is the surface area bounded by the path of integration of $\vec{H}$, we can write $$\oint\vec{H}.\vec{dl}=\int_S\vec{J}.\vec{ds}$$

Using stokes theorem, this can be written as

$$\oint_S(\vec{\nabla}\times\vec{H}.\vec{ds}=\int_S\vec{J}.\vec{ds}$$

Hence, $\vec{\nabla}\times\vec{H}=\vec{J}$

Since,$\vec{B}=\mu_0\vec{H}$ in free space, we can write

$$ \vec{\nabla}\times\vec{B}=\mu_0\vec{J}$$

This is called the differential or point form of ampere’s circuital law.