1. Interpret the geometry

   Two infinitely long straight wires lie in the $xy$-plane and are mutually perpendicular.

   • One wire is along the $x$-axis.
   • The other wire is along the $y$-axis.
   • Point $P$ is at equal perpendicular distance $d$ from both wires, i.e. effectively at $(d,d,0)$ in the plane.

   The currents have equal magnitude $I$, with directions as shown.

2. Magnetic field due to a long straight wire

   For an infinite straight wire, the magnitude of magnetic field at perpendicular distance $r$ is

   $$B = \frac{\mu_0 I}{2\pi r}.$$

   Here, for both wires, the distance of point $P$ from the wire is $d$. Hence each wire produces field of magnitude

   $$B_0 = \frac{\mu_0 I}{2\pi d}.$$

3. Direction of field due to the wire along the $x$-axis

   Take the wire along the $x$-axis carrying current in the shown direction. Using the right-hand thumb rule:

   • At a point above the $x$-axis in the $xy$-plane ($y>0$), the magnetic field is along $+\hat z$ or $-\hat z$ depending on current direction.

   From the figure’s current directions, the field at $P$ due to this wire is along one $z$-direction, say

   $$\vec B_x = +\frac{\mu_0 I}{2\pi d}\,\hat z.$$

4. Direction of field due to the wire along the $y$-axis

   Similarly, applying the right-hand rule to the wire along the $y$-axis, at the point $P$ (to the right of the $y$-axis, $x>0$), the field is in the opposite $z$-direction:

   $$\vec B_y = -\frac{\mu_0 I}{2\pi d}\,\hat z.$$

5. Net magnetic field

   Since magnitudes are equal and directions are opposite,

   $$\vec B_{\text{net}} = \vec B_x + \vec B_y = \frac{\mu_0 I}{2\pi d}\hat z - \frac{\mu_0 I}{2\pi d}\hat z = 0.$$

6. Check options

   • A: nonzero along $\hat z$ $\rightarrow$ incorrect
   • B: nonzero in $xy$-plane $\rightarrow$ impossible here, incorrect
   • C: Zero $\rightarrow$ correct
   • D: nonzero in $xy$-plane $\rightarrow$ incorrect

Therefore, the net magnetic field at $P$ is