Explain Biot-Savart law for the magnetic field produced at a point due to the current flowing in a current element. How will you determine the direction of the field?

Explanation

Biot-Savart.s Law : The small magnetic field produced at a given observation point le to current flowing through the elementary portion of the wire is :
(i) directly proportional to the current i.e.,
`dB prop I`
(ii) directly proportional to the length of the elementary portion i.e.,
`dB prop dl`
(iii) directly proportional to sine of the angle between the direction of the flow of current and the line joining the elementary portion to the observation point i.e.,
`dB prop sin theta`
(iv) inversely proportional to the square of the distance r of the point P from the clementary portion i.e.,
`dB prop 1/(r^2)`
Combining all the above factors, we have
`dB prop (I dl sin theta)/(r^2)`
`implies dB = k (I dl sin theta)/(r^2)`
But `k = (mu_0)/(4pi)`= Constant of proportionality
`= 10^(-7) wb m^(-1) A^(-1) " " `[In S.I. System]
and `mu_0 =` Absolute permeability of free space.
In CGS, `k = 1`
So `dB = (mu_0)/(4pi) (I dl sin theta)/(r^2)`
In vector form, `dvecB = (mu_0)/(4pi) (I vec(dl) xx vec(r))/(r^2)`
where `hatr =` a unit vector along CP.
But `hatr = (vecr)/(r)`
`dvecB = (mu_0)/(4pi) (I vec(dl) xx vecr)/(r^3)`
Direction of magnetic field at P: Direction of magnetic field at point P is given by using Right Hand Thumb Rule.
According to this rule, magnetic field `(vec(dB))` at point P is perpendicular to the plane containing `vec(dl)` and `vecr` and is directed downward in the plane of paper. It is represented by X.