fantastic problem

The question is simple but looks intimidating......

There is a method to solve it by using ampere's law..

But the more rigorous way (probably gives more insight into this problem is)

By breaking the whole plane into lot of thin wires of thickness dt and then using the current 2adt ....

If the point above the axis is at a distance y,

then you need to integrate this value.

Note that if the two wires are taken at an equal distance t on each side of the point on the plane, their y component will get cancelled out!!

The x component will be something of the form

2.μ₀2adt/{2.Ï€.√y²+t²} . y/√y²+t²

integrate this from 0 to ∞

this will be a simple expression which will be

2aμ₀/Ï€ x (Ï€/2)

= μ₀a