Two identical long conducting wires AOB and Cod are placed at right angles to each other, with one above other such that O is their common point for the two. The wires carry `I_(1) and I_(2)` currents, respectively. Points P is lying at distance d form O along a direction perpendicular to the plane containing the wires. The magnetic field at the point P will be

Options

A. `mu_(0)/(2 pi d) (I_(1)/I_(2))`
B. `mu_(0)/(2 pi d) (I_(1) + I_(2))`
C. `mu_(0)/(2 pi d) (I_(1)^(2) - I_(2)^(2))`
D. `mu_(0)/(2 pi d) (I_(1)^(2) + I_(2)^(2))^(1//2)`

Answer

d

Explanation

Let the wires be placed Y as shown in the figure below such that one of the wires is carrying current , along X-axis and another is carrying current along Y-axis. Also, the point P (0, 0, d) is taken on the Z-axis, perpendicular to the X-Y plane.
Magnetic field at P due to currently `I_(1) " in wire" AOB " is " B_(1) = (mu_(0) I_(1))/(2 pi d) ` , , along Y-axis
Magnetic field at P due to current `I_(2) " in wire " COD " is " B_(2) = (mu_(0)I_(2))/(2 pi d) `, along X-axis
The magnetic field `B_(1)" and "B_(2)` are mutually perpendicular to each other at point P.
Net magnetic field at P will be
`B_("net") = sqrt(B_(1)^(2) + B_(2)^(2)) = mu_(0)/(2 pi d) sqrt(I_(1)^(2) + I_(2)^(2)) `