A magnetic field existing in a region is given by B^-=B0[1+(x / ℓ)] k ∧. A square loop of side ℓ

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A magnetic field existing in a region is given by...

A magnetic field existing in a region is given by $\mathrm{B}^{-}=\mathrm{B}_{0}[1+(\mathrm{x} / \ell)] \mathrm{k} \wedge . \mathrm{A}$ square loop of side $\ell$ and carrying current I is placed with edges (sides) parallel to $\mathrm{X}-\mathrm{Y}$ axis. The magnitude of the net magnetic force experienced by the Loop is (a) $2 \mathrm{~B}_{0} \overline{\mathrm{I} \ell}$ (b) $(1 / 2) B_{0} I \ell$ (c) $\mathrm{B}_{\circ} \mathrm{I} \ell$ (d) BI\ell

A magnetic field existing in a region is given by $\mathrm{B}^{-}=\mathrm{B}_{0}[1+(\mathrm{x} / \ell)] \mathrm{k} \wedge . \mathrm{A}$ square loop of side $\ell$ and carrying
current I is placed with edges (sides) parallel to $\mathrm{X}-\mathrm{Y}$ axis. The magnitude of the net magnetic force experienced by the Loop is
(a) $2 \mathrm{~B}_{0} \overline{\mathrm{I} \ell}$
(b) $(1 / 2) B_{0} I \ell$
(c) $\mathrm{B}_{\circ} \mathrm{I} \ell$
(d) BI\ell

Key Concepts

Magnetic Dipole and Torque

Although the focus here is on net magnetic force, the concept of a magnetic dipole (a current loop) in a magnetic field is important. In uniform fields, a current loop may not experience a net force but rather a torque. In non-uniform fields, however, both net forces and torques can arise. This concept helps in understanding the broader behavior of current loops in varying magnetic field environments.

Integration of Differential Forces

When a magnetic field is non-uniform, the force acting on a current element may vary along its length. In such cases, it is necessary to integrate these differential force contributions over the entire path of the conductor to obtain the net force on the loop. This process is critical in solving problems where the field is not constant.

Non-uniform Magnetic Field

This concept refers to magnetic fields whose magnitude or direction changes with position. In problems involving spatially varying fields, different segments of a current loop experience different levels of force, leading to a net force or torque on the loop. Understanding this helps in analyzing how the variation in B affects the overall behavior of a current-carrying conductor.

Lorentz Force Law

The Lorentz force law is fundamental to electromagnetism and describes the force acting on a charge moving in a magnetic field. For a current-carrying conductor, it states that the force is given by F = I (L × B), where I is the current, L is the length vector along the current direction, and B is the magnetic field. This principle is used to determine the force on each segment of the loop.

Force on a Current-Carrying Conductor

A current-carrying conductor in a magnetic field experiences a force due to the interaction between the current and the magnetic field. The magnitude of this force on a small segment is proportional to the current, the length of the segment, and the local magnetic field, with the direction given by the right-hand rule. This is a key concept when calculating net forces on different segments of a loop.

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