Two straight long conductors A O B and C O D are...

Two straight long conductors $A O B$ and $C O D$ are perpendicular to each other and carry currents $i_{1}$ and $i_{2}$. The magnitude of the magnetic induction at a point $P$ at a distance $a$ from the point $O$ in a direction perpendicular to the plane $A C B D$ is (A) $\frac{\mu_{0}}{2 \pi a}\left(i_{1}+i_{2}\right)$ (B) $\frac{\mu_{0}}{2 \pi a}\left(i_{1}-i_{2}\right)$ (C) $\frac{\mu_{0}}{2 \pi a}\left(i_{1}^{2}+i_{2}^{2}\right)^{1 / 2}$ (D) $\frac{\mu_{0}}{2 \pi a} \frac{i_{1} i_{2}}{\left(i_{1}+i_{2}\right)}$

Two straight long conductors $A O B$ and $C O D$ are perpendicular to each other and carry currents $i_{1}$ and $i_{2}$. The magnitude of the magnetic induction at a point $P$ at a distance $a$ from the point $O$ in a direction perpendicular to the plane $A C B D$ is
(A) $\frac{\mu_{0}}{2 \pi a}\left(i_{1}+i_{2}\right)$
(B) $\frac{\mu_{0}}{2 \pi a}\left(i_{1}-i_{2}\right)$
(C) $\frac{\mu_{0}}{2 \pi a}\left(i_{1}^{2}+i_{2}^{2}\right)^{1 / 2}$
(D) $\frac{\mu_{0}}{2 \pi a} \frac{i_{1} i_{2}}{\left(i_{1}+i_{2}\right)}$

Key Concepts

Superposition Principle in Magnetostatics

The superposition principle states that in magnetostatics, the total magnetic field produced by multiple sources is the vector sum of the fields produced by each individual source. This principle is critical when dealing with problems involving several conductors, as it allows the individual magnetic fields to be calculated independently and then combined to determine the net magnetic induction at a given point.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field generated by a current-carrying conductor. By pointing the thumb of the right hand in the direction of the conventional current, the curled fingers show the direction in which the magnetic field lines wrap around the conductor. This rule is essential for correctly assigning the direction to each field before applying the superposition principle.

Magnetic Field of a Long Straight Conductor

This concept refers to the determination of the magnetic field generated by a long, straight current-carrying conductor. The field at any point at a perpendicular distance r from the wire is given by the expression B = (??i) / (2?r), which is derived from Ampère’s law or the Biot–Savart law. This formula forms the foundation for analyzing the contribution of each individual conductor to the overall magnetic field in magnetostatic problems.

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