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)   \begin{equation*} \oint \boldsymbol{H}.\boldsymbol{dl} = I \end{equation*}

where
I \rightarrow total current passing through interior of the path
H \rightarrow magnetic field intensity parallel to the path

Long straight conductor carrying current 'I'

Long straight conductor carrying current ‘I’

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, \oint \boldsymbol{H}.\boldsymbol{dl} = \mu I_r
where, I_r is the current enclosed the loop at radius ‘r’ from the center of conductor.

Fraction of total current that is indise the loop,

(2)   \begin{eqnarray*} {I_r \over 4\pi r^2} = {I \over 4\pi R^2} \Rightarrow I_r = I \left({r \over R}\right)^2 \end{eqnarray*}

Inside conductor (r<R)  Outside conductor (r>R)
Magnetic field intensity (\boldsymbol{H}_r) at a distance r(<R) from the center of conductor,

(3)   \begin{eqnarray*} \oint \boldsymbol{H_r}.\boldsymbol{dl} &= \mu I_r \\ \Rightarrow ~~ H_r ~ 2\pi r &= \mu I \left({r \over R}\right)^2\end{eqnarray*}

So Magnetic field intensity (\boldsymbol{H}_r) inside the conductor

(4)   \begin{equation*} \boxed{H_r = {\mu I \over 2\pi} {r \over R^2}} \end{equation*}

Magnetic field intensity (\boldsymbol{H}_r) at a distance r(>R) from the center of conductor,

(5)   \begin{eqnarray*} \oint \boldsymbol{H_r}.\boldsymbol{dl} &= I \\ \Rightarrow ~~ H_r ~ 2\pi r &= \mu I \end{eqnarray*}

So Magnetic field intensity (\boldsymbol{H}_r) outside the conductor

(6)   \begin{equation*} \boxed{H_r = {\mu I \over 2\pi r} } \end{equation*}

 

 

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)   \begin{equation*} \oint \boldsymbol{H}.\boldsymbol{dl} &= \mu NI & ~~~~ \Rightarrow ~~ H ~ 2\pi R = \mu NI \end{equation*}

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

(8)   \begin{equation*} \boxed{H = { \mu NI \over 2\pi R}} \end{equation*}

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