# Ampere’s Law

Ampere’s law relates the integrated magnetic field around a closed loop to the electric current passing through the surface of the loop.  It was first discovered  by André-Marie Ampère in 1826, later re-derived by Maxwell using hydrodynamics in 1861. Now it is one of the Maxwell equations, which is the basis for classical electro-magnetism.

## Ampere’s Law – Definition

Ampere’s law states that “the total magneto motive force (MMF) around a closed path is equal to the total current passing through the interior of the closed path
Mathematically,

(1) where
I total current passing through interior of the path
H magnetic field intensity parallel to the path

## Ampere’s Law Applications

– Magnetic field from a infinite long straight conductor
– Magnetic field inside a conductor
– Magnetic field inside a long solenoid and torodial coil

### Magnetic field inside and outside of a infinite long straight conductor

Consider a long conductor of radius ‘R’, carrying current ‘I’ uniformly distributed across the conductor.
By Ampere’s Law, where, is the current enclosed the loop at radius ‘r’ from the center of conductor.

Fraction of total current that is indise the loop,

(2) Inside conductor (rR) Magnetic field intensity ( ) at a distance r( ) from the center of conductor, (3) So Magnetic field intensity ( ) inside the conductor (4) Magnetic field intensity ( ) at a distance r( ) from the center of conductor, (5) So Magnetic field intensity ( ) outside the conductor (6) ### Magnetic field inside a toroid coil

Consider a toroid of radius ‘R’ and having ‘N’ number of turns.
The current enclosed by the dashed line is N times the current in each loop.
By Ampere’s law,

(7) From Eq-(7), field intensity inside a toroid coil is

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