This post covers Ampere’s Circuital Law with its statement and formula.
Ampere’s Circuital Law in electromagnetism is analogous to Gauss’ Law in electrostatics.
Ampere’s Circuital Law – statement
Ampere’s Circuital Law states that the line integral of the magnetic field B around any ‘closed’ path is equal to μ0 times the net charge I passing through the area enclosed by the path.
Ampere’s Circuital Law – formula
The formula presenting Ampere’s Circuital Law is as follows:
ʃL B. dL = μ0 IAmpere’s Circuital Law
Here μ0 is the permeability of the free space.
[ Also read: Biot-Savart Law statement, derivation, formula]
Ampere’s Circuital Law from Biot-Savart law (derive or obtain)
Ampere’s circuital law can be described as Biot-Savart law expressed in an alternative way.
Here, we will briefly derive or obtain Ampere’s circuital law from the formula of the magnetic field at a point P due to an infinitely long straight current-carrying conductor (this formula is derived using the Biot-Savart law). So it may be said that we will obtain Ampere’s Circuital Law from Biot – Savart law.
The magnetic field at a point P due to an infinitely long straight current carrying conductor is B = μ0 I / (2πr), where P is a point at a distance r from the conductor.
B (2πr) = μ0 I
B (2πr) is the product of the magnetic field and the circumference of the circle of radius ‘r’ on which the magnetic field is constant.
If L is the perimeter of the closed curve and I is the net current enclosed by the closed curve, then the above equation may be expressed as,
BL = μ0I
In a more generalized way, Ampere’s circuital law is written as ʃL B. dL = μ0 I
The line integral does not depend on the shape of the path or the position of the wire within the magnetic field. If the current in the wire is in the opposite direction, the integral would have the opposite sign.
If the closed path does not encircle the wire (if a wire lies outside the path), the line integral of the field of that wire is zero.
Although derived for the case of a number of long straight parallel conductors, the law is true for conductors and paths of any shape.