Two semi infinitely long straight current carrying conductors are held in the form as shown in figure. One common end of them is at the origin. If both the conductors carry same current I, find the value of the magnetic field induction at a point (a, b).

Explanation

Magnetic field induction at point P due to current through the conductor along X-axis is
`B_1=(mu_0)/(4pi)I/b[sin theta_1+sin pi/2]`
acting along negative Z-axis.
`=(mu_0)/(4pi)I/b[(a)/(sqrt(a^2+b^2))+1]`
Magnetic field induction at point P due to current through the conductor along Y-axis is
`B_2=(mu_0)/(4pi)I/a[sin theta_2+sin pi/2]`
acting along negative Z-axis.
`=(mu_0)/(4pi)I/a[(b)/(sqrt(a^2+b^2))+1]`
The resultant magnetic field induction at P is
`B=B_1+B_2` acting along negative Z-axis.
`=(mu_0)/(4pi)I/b[(a)/(sqrt(a^2+b^2))+1]+(mu_0)/(4pi)I/a[(b)/(sqrt(a^2+b^2))+1]`
`=(mu_0I)/(4pi)[(1/a+1/b)+(1)/(sqrt(a^2+b^2))(a/b+b/a)]`
or `B=(mu_0I)/(4pi)[((a+b))/(ab)+(sqrt(a^2+b^2))/(ab)]`
`=(mu_0I)/(4piab)[(a+b)+sqrt(a^2+b^2)]`