State and explain Biot-Savart law for the magnetic field produced by a current element. Give the direction of magnetic field and define the unit of it.

Explanation

1. Biot-Savart Law : Magnetic field of position vector of point relative to element "I`dvecl` which is current carrying element at `vecr` position vector is given by `dvecB=mu_(0)/(4pi)*(Idveclxxvecr)/r^(2)`.

2. According to Biot-Savart law, the magnitude of the field is `dvecB`.
(1) Directly proportional to the current I through the conductor,
`thereforedBpropI`
(2) Directly proportional to the length `|dvecl|` of the current element,
`thereforedBpropdl`
(3) Directly proportional to `sintheta`,
`thereforedBpropsintheta`
(4) Inversely proportional to the square of the distance r of the point P from the current element,
`thereforedBprop1/r^(2)`
Combining all these four factors, we get
`dBprop(Idlsintheta)/r^(2)`
`dBprop(Idlrsintheta)/r^(3)`
`dvecBprop(Idveclxxvecr)/r^(3)`
`thereforedvecB=mu_(0)/(4pi)(Idveclxxvecr)/r^(3)" "...(1)`
3. Where `mu_(0)/(4pi)` is proportionally constant.
4. Where `mu_(0)` is vacuum permeability.
It.s value is `mu_(0)=4pixx10^(-7)(T*m)/A`
`therefore(mu_(0))/(4pi)=10^(-7)(T*m)/AorNA^(-2)`
5. Equation (1) is for magnetic field in vacuum.
6. Magnitude of electric field from equation (1) is given by,
`|dvecB|=mu_(0)/(4pi)(Idlrsintheta)/r^(3)ormu_(0)/(4pi)*(|Idveclxxvecr|)/r^(3)`
where `theta` is the angle between `Idveclandvecr`.
`therefore|dvecB|=mu_(0)/(4pi)(Idlsintheta)/r^(2)" "...(2)`
7. The direction of `dvecB` is the direction of the perpendicular to plane of cross product of `Idveclandvecr` which is inside to plane of paper which is indicated as
near point P.