Question

State and explain Ampere's circuital law.

Answer 1

Amperes circulate law: The line integral of magnetic field of induction $\overrightarrow{B}$ around any closed path in free space is equal to absolute permeability of free space ${\mu }_0$ times the total current flowing through area bounded by the path.
Mathematically, $\varphi \overrightarrow{B}\cdot \overrightarrow{dl}={\mu }_{0}I$
where B-magnetic induction, I-Total current, dl-length element of path, ${\mu }_0$-Permeability of the space.
Explanation:
(a) Ampere's law is generalisation of Biot-Savart's law and is used to determine magnetic field at any point due to distribution of current.
(b) Consider a long straight current carrying conductor XY, placed in vacuum. A steady current 'I' flows through it from the end Y to X as shown in the figure.
(c) Imagine a closed curve(amperian loop) around the conductor having radius 'r'.
(d) The loop is assumed to be made of large number of small elements each of length $\overrightarrow{dl}$.
(e) Its direction is along the direction of traced loop.
(f) Let $\overrightarrow{B}$ be the strength of magnetic field around the conductor.
(g) All the scalar product of $\overrightarrow{B}$ and $\overrightarrow{dl}$ give the product of ${\mu }_0$ and I.
It is given by $\varphi =\overrightarrow{B}\cdot \overrightarrow{dl}=$ $Bdl$ $\cos \theta$

where, $\theta$ =angle between $\overrightarrow{B}$ and $\overrightarrow{dl}$