Derive equation of missing term in Ampere circuital law. Write its definition and unit.

Explanation

1. There is an alternative and appealing way in which the Biot-Savart.s law may be expressed.
2. As shown in figure, Ampere.s circuital law considers an open surface with a boundary.

3. The surface has current passing through it. We consider the boundary to be made up of a number of small line elements. Consider one such element of length dl.
4. We take the value of the tangential component of the magnetic field `B_(T)` at this element and multiply it by the length of that element dl,
`B_(T)dl=vecB*vec(dl)`
= `B_(T)dlcos0^(@)`
= `B_(T)dl`
5. The sum then tends to an integral.
6. Ampere.s circuital law : The line integral of magnetic induction over a closed loop in a magnetic field is equal to the product of algebric sum of electric currents enclosed by the loop and the magnetic permeability.
`thereforeointvecB*vec(dl)=mu_(0)sumI`
7. Where `sumI` is the total current through the surface.
8. Let L be the length of the loop for which `vecB` is tangential and `I_(e)` be the current enclosed by the loop.
`thereforeointvecB*vec(dl)=BLandsumI=I_(e)`
`therefore` According to Ampere.s circuital law BL = `mu_(0)I_(e)`
9. The boundary of the loop chosen is a circle and magnetic field is tangential to the circumference of the circle, then Ampere.s law,
`Bxx2pir=mu_(0)I`
`thereforeB=(mu_(0)I)/(2pir)`
10. The Ampere.s law involves a sign-convention given by the right hand rule.
11. Let the fingers of the right hand be curled in the sense the boundary is traversed in the loop integral `ointvecB*dvecl`.
12. Then the direction of the thumb gives the sense in which the current I is regarded as positive.