1. Interpret the geometry

The two identical conducting wires $AOB$ and $COD$ are mutually perpendicular and lie in the same plane.

• Let wire $AOB$ lie along the $x$-axis carrying current $I_1$.
• Let wire $COD$ lie along the $y$-axis carrying current $I_2$.
• The observation point is at distance $d$ from $O$ on the line perpendicular to the plane of the wires, say along the $z$-axis.

So the point is at $P(0,0,d)$.

Because each wire is effectively an infinite straight conductor (with $O$ as their crossing point), the perpendicular distance from point $P$ to each wire is $d$.

2. Magnetic field due to each wire

For a long straight wire, the magnetic field magnitude at perpendicular distance $d$ is

Therefore,

• Field due to wire $AOB$:

  $$B_1 = \frac{\mu_0 I_1}{2\pi d}$$

• Field due to wire $COD$:

  $$B_2 = \frac{\mu_0 I_2}{2\pi d}$$

3. Direction of the two magnetic fields

Using the right-hand rule:

• For the wire along the $x$-axis, at point on the $z$-axis, the magnetic field is along the $y$-direction.
• For the wire along the $y$-axis, at the same point, the magnetic field is along the $x$-direction.

Hence, the two magnetic field vectors are perpendicular to each other.

4. Resultant magnetic field

Since $\vec B_1 \perp \vec B_2$,

Substitute the values:

Factor out $\frac{\mu_0}{2\pi d}$:

5. Match with options

This corresponds to:

So the correct option is C.

6. Comparison with stored correct answer

Stored correct answer: C

My derived answer: C

They agree.