According to the sources, the magnetic field $B$ produced by a long straight wire carrying current $I$ at a distance $d$ is given by $B = \frac{\mu_0 I}{2\pi d}$ .

1. Field from Wire 1 ($B_1$): For the wire carrying current $I_1$, the magnetic field at point $P$ (at a perpendicular distance $d$) is $B_1 = \frac{\mu_0 I_1}{2\pi d}$. Using the right-hand rule, this field vector lies in the plane of the wires, perpendicular to the first wire.
2. Field from Wire 2 ($B_2$): Similarly, for the wire carrying current $I_2$, the magnetic field at point $P$ is $B_2 = \frac{\mu_0 I_2}{2\pi d}$. This field vector also lies in the plane of the wires but is perpendicular to the second wire.
3. Resultant Field ($B$): Since the two wires are at right angles, the magnetic field vectors $\vec{B}_1$ and $\vec{B}_2$ they produce at point $P$ are also perpendicular to each other. The magnitude of the net magnetic field is calculated using the Pythagorean theorem: $B = \sqrt{B_1^2 + B_2^2}$.

Substituting the values: $B = \sqrt{(\frac{\mu_0 I_1}{2\pi d})^2 + (\frac{\mu_0 I_2}{2\pi d})^2} = \frac{\mu_0}{2\pi d} \sqrt{I_1^2 + I_2^2}$, which can be written as $\frac{\mu_0}{2\pi d} (I_1^2 + I_2^2)^{1/2}$.