Resultant Field Of Two Wires

Each long straight wire produces a field of magnitude $B_0=\frac{{\mu }_{0}I}{2\pi r}$ at the point in question, since both are a distance $r$ away. The key step is recognizing that magnetic field is a vector, so the two contributions must be added vectorially, not arithmetically. Given that the two field vectors are mutually perpendicular at that point, the resultant magnitude is $B=\sqrt{B_0^2+B_0^2}=\sqrt{2}{B}_0=\frac{\sqrt{2}{\mu }_{0}I}{2\pi r}$. The option $\frac{{\mu }_{0}I}{2\pi r}$ ignores the second wire's contribution entirely. The option $\frac{2{\mu }_{0}I}{2\pi r}$ wrongly adds the magnitudes as if the fields were parallel. The option $0$ would require the fields to be antiparallel and equal, which is not the geometry described. This superposition principle for magnetic fields is emphasized throughout the NCERT treatment of the Biot-Savart law. A sanity check confirms that for two equal perpendicular vectors the resultant exceeds either one by the factor $\sqrt{2}\approx 1.41$, which is physically reasonable. This vector superposition is the magnetic counterpart of how electric fields from several charges combine, and forgetting the directional nature of the field is one of the most common errors in problems with multiple conductors. In configurations where the two currents flow so that their fields oppose, the same construction can yield a much smaller or even zero resultant, so identifying the relative directions before adding magnitudes is always the crucial first step.