Ampere's circuital law in magnetism is analogous to gauss's law in electrostatics
This law is also used to calculate the magnetic field due to any given current distribution
This law states that
" The line integral of resultant magnetic field along a closed plane curve is equal to μ_{0} time the total current crossing the area bounded by the closed curve provided the electric field inside the loop remains constant" Thus
where μ_{0} is the permeability of free space and I_{enc} is the net current enclosed by the loop as shown below in the figure
The circular sign in equation (21) means that scalar product B.dl is to be integrated around the closed loop known as Amperian loop whose beginning and end point are same
Anticlockwise direction of integration as chosen in figure 9 is an arbitrary one we can also use clockwise direction of integration for our calculation depending on our convenience
To apply the ampere's law we divide the loop into infinitesimal segments dl and for each segment, we then calculate the scalar product of B and dl
B in general varies from point to point so we must use B at each location of dl
Amperian Loop is usually an imaginary loop or curve ,which is constructed to permit the application of ampere's law to a specific situation
Proof Of Ampere's Law
Consider a long straight conductor carrying current I perpendicular to the page in upward direction as shown below in the figure
From Biot Savart law, the magnetic field at any point P which is at a distance R from the conductor is given by
Direction of magnetic Field at point P is along the tangent to the circle of radius R withTh conductor at the center of the circle
For every point on the circle magnetic field has same magnitude as given by
And field is tangent to the circle at each point
The line integral of B around the circle is
since ∫dl=2πR ie, circumference of the circle so,
This is the same result as stated by Ampere law
This ampere's law is true for any assembly of currents and for any closed curve though we have proved the result using a circular Amperian loop
If the wire lies outside the Amperian loop, the line integral of the field of that wire will be zero
but does not necessarily mean that B=0 everywhere along the path ,but only that no current is linked by the path
while choosing the path for integration ,we must keep in mind that point at which field is to be determined must lie on the path and the path must have enough symmetry so that the integral can be evaluated