Derive an expression for magnetic field induction on the equatorial line of a barmagnet.

Explanation

At a point on equatorial line : Let us consider a point 'P' at a distance 'd' on the equatorial line from the centre of a bar magnet.
`B_(N) = (mu_(0))/(4pi)(m xx l)/((sqrt(d^(2) + l^(2)))) = (mu_(0))/(4pi)(m)/(d^(2) + l^(2))` and `B_(S) = (m xx l)/((sqrt(d^(2)+l^(2)))) = (mu_(0))/(4pi)(m)/(d^(2)+l^(2))`
`B_("equ") = B_(N)cos theta + B_(N) cos theta = 2B_(N) cos theta`
`= (2mu_(0) m)/(4pi (d^(2) + l^(2)))(1)/(sqrt(d^(2)+l^(2))) = (mu_(0))/(4pi)(2lm)/((sqrt(d^(2) + l^(2)))^((3)/(2))) :. B_("equ") = (mu_(0))/(4pi)(M)/((sqrt(d^(2)+l^(2)))^((3)/(2)))`
Special case : In case of short bar magnet, `B_("equ") = (mu_(0))/(4pi)(M)/(d^(3))bar(NS)` (as `l lt lt d, d^(2) + l^(2) ~~ d^(2)`)