Two straight long conductors AOB and COD are perpendicular to each other and carry currents I1 a

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Two straight long conductors AOB and COD are perpendicular...

Two straight long conductors $\mathrm{AOB}$ and $\mathrm{COD}$ are perpendicular to each other and carry currents $\mathrm{I}_{1}$ and $\mathrm{I}_{2}$. The magnitude of the mag. field at a point " $\mathrm{P}^{n}$ at a distance " $\mathrm{a}^{\prime \prime}$ from the point "O" in a direction perpendicular to the plane $\mathrm{ABCD}$ is (a) $\left[\left(\mu_{0}\right) /(2 \pi a)\right]\left(I_{1}+I_{2}\right)$ (b) $\left[\left(\mu_{0}\right) /(2 \pi a)\right]\left(I_{1}-I_{2}\right)$ (c) $\left[\left(\mu_{0}\right) /(2 \pi a)\right]\left(\mathrm{I}_{1}^{2}+\mathrm{I}_{2}^{2}\right)^{1 / 2}$ (d) $\left[\left(\mu_{0}\right) /(2 \pi a)\right]\left[\left(I_{1} I_{2}\right) /\left(I_{1}-I_{2}\right)\right]$

Two straight long conductors $\mathrm{AOB}$ and $\mathrm{COD}$ are perpendicular to each other and carry currents $\mathrm{I}_{1}$ and $\mathrm{I}_{2}$. The magnitude of the mag. field at a point " $\mathrm{P}^{n}$ at a distance " $\mathrm{a}^{\prime \prime}$ from the point "O" in a direction perpendicular to the plane $\mathrm{ABCD}$ is
(a) $\left[\left(\mu_{0}\right) /(2 \pi a)\right]\left(I_{1}+I_{2}\right)$
(b) $\left[\left(\mu_{0}\right) /(2 \pi a)\right]\left(I_{1}-I_{2}\right)$
(c) $\left[\left(\mu_{0}\right) /(2 \pi a)\right]\left(\mathrm{I}_{1}^{2}+\mathrm{I}_{2}^{2}\right)^{1 / 2}$
(d) $\left[\left(\mu_{0}\right) /(2 \pi a)\right]\left[\left(I_{1} I_{2}\right) /\left(I_{1}-I_{2}\right)\right]$

Key Concepts

Magnetic Field due to a Long Straight Conductor

This concept explains how a steady electric current flowing through a long, straight conductor generates a magnetic field in the surrounding space. Its magnitude is typically given by an expression derived using either the Biot–Savart Law or Ampere’s Law, showing that the magnetic field decreases inversely with the distance from the conductor.

Biot–Savart Law

The Biot–Savart Law provides a fundamental means of calculating the magnetic field produced by a small segment of current-carrying conductor. It relates the magnetic field contribution to the current element, the distance from the element to the point of interest, and the sine of the angle between the current direction and the position vector, forming the basis for deriving the magnetic effect of more complex current distributions.

Superposition Principle in Electromagnetism

This principle states that when multiple magnetic fields are present, the net magnetic field at any point is the vector sum of the individual fields produced by each current element or source independently. It is crucial for analyzing systems where multiple conductors, such as two perpendicular wires, contribute magnetic fields in overlapping regions.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field around a current-carrying conductor. By aligning the thumb in the direction of conventional current, the way the fingers curl represents the magnetic field lines, an essential method for establishing the orientation of the magnetic field in three-dimensional problems.

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