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 $A B C D$, 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) $\frac{\mu_{0}}{2 \pi a}\left(\hat{r}_{1}+\hat{z}_{2}\right)^{1 / 2}$ (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 $A B C D$, 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) $\frac{\mu_{0}}{2 \pi a}\left(\hat{r}_{1}+\hat{z}_{2}\right)^{1 / 2}$
(d) $\frac{\mu_{0}}{2 \pi a} \frac{i_{1} i_{2}}{\left(i_{1}+i_{2}\right)}$

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Key Concepts

Magnetic Field of a Long Straight Conductor

A long straight conductor carrying a current produces a magnetic field that circulates around the wire. The magnitude of the field at any point at a distance r from the wire is given by an expression proportional to the current and inversely proportional to the distance, and is derived using Ampere's law or the Biot–Savart law.

Superposition Principle

When multiple magnetic fields are present, as in the case of two or more current-carrying conductors, the net magnetic field at any point is the vector sum of the individual fields. This principle allows for independent calculation of each field contribution, which are then combined to determine the overall magnetic induction.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field produced by a current-carrying conductor. It relates the direction of the current to the orientation of the magnetic field lines, thus aiding in the correct vector addition when multiple fields are involved.

Magnetic Induction

Magnetic induction, also known as the magnetic field, quantifies the magnitude and direction of the magnetic influence produced by currents. Its calculation in complex configurations involves both understanding the individual contributions and employing vector addition to obtain the net effect.

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