Write expression for the magnetic flux density at a point due to a current carrying wire of:
a finite length?

Explanation

Consider a long straight wire LM.
Let I = Current flowing through the wire from L to M.
Let P be the observation point, where we want to calculate the magnetic field.
OP = a = Perpendicular distance of P from the current carrying wire
Consider the small element CD of length dl of the wire and I is distance of mid-point of elementary portion CD from O.
`r = OʻP` = Distance of mid-point of small elementary portion CD from P.

0 = Angle between the current carrying wire and the line joining the mid-point of CD to P, i.e., between `vec(dl)` and `vecr` .
According to Biot-Savart.s law, magnetic induction `vec(dB)` at the point P due to small current element CD is given by
`vec(dB) = (mu_0)/(4pi) (I vec(dl) xx hatr)/(r^2)`
In magnitude `dB = (mu_0)/(4pi)(I dl sin theta)/(r^2) " " ...(1)`
and `sin theta = d/r = cos theta " " .....(2)`
or `r = (a)/(cos phi) = a sec phi " " ....(3)`
Also , `tan phi = l//a or l = a tan phi" " ....(4)`
`dl = a sec^2 phi d phi`
Substituting the values from eqns. (2), (3) and (4) in eqn. (1), we get
`dB = (mu_0)/(4pi) (I(a sec^2 phi d phi)cos phi)/(a^2 sec^2 phi)`
`dB = (mu_0 I)/(4pi a) cos phi d phi " " ....(5)`
Join PM and PL. Let `/_OPM.= phi_2` and `/_OPL = - phi_1`
Then, magnetic field due to the whole length of conductor LM can be calculated by integrating eqn. (5) within the limits `-phi_1` to `-phi_2`. Thus,
`B = int_(-phi_1)^(phi_2) dB = (mu_0 I)/(4pi a) int_(-phi_1)^(phi_2) cos phi d phi`
`= (mu_0 I)/(4pi a) [sin phi]_(-theta_1)^(-theta_2) = (mu_0I)/(4pi a) [ sin phi_2 - sin(phi_1)]`
`= (mu_0 I)/(4pi a)[sin phi_2 + sin phi_1] " " ....(6)`
The above expression gives the magnitude of the magnetic field due to current carrying straight conductor of finite length.
If the conductor is infinitely long and the point P lies near the centre of the conductor, then
`phi_1 = phi_2 = pi/2`
So, `B = (mu_0 I)/(4pi a)["sin" pi/2 + "sin" pi/2]`
`B = (mu_0 2I)/(4 pi a)`
It is the expression for magnetic field due to an infinitely long current carrying straight conductor.