A conducting rod of mass m and length l is connected by two...

A conducting rod of mass $m$ and length $l$ is connected by two identical springs as shown in Fig. $15.47$. Initially, the system is in equilibrium. A uniform magnetic field of magnitude $B$ directed perpendicular to the plane of the paper outwards also exists in the region. If a current $I$ is switched on that passes from $P$ to $Q$ through the rod. Further maximum elongation in the spring is $[$ Given: $|m g|=|B I|]$ Fig. (A) $\frac{B I l}{K}$ (B) $\frac{B I l}{4 K}$ (C) $\frac{B I l}{8 K}$ (D) $\frac{B I l}{16 K}$

A conducting rod of mass $m$ and length $l$ is connected by two identical springs as shown in Fig. $15.47$. Initially, the system is in equilibrium. A uniform magnetic field of magnitude $B$ directed perpendicular
to the plane of the paper outwards also exists in the region. If a current $I$ is switched on that passes from $P$ to $Q$ through the rod. Further maximum elongation in the spring is $[$ Given: $|m g|=|B I|]$
Fig.
(A) $\frac{B I l}{K}$
(B) $\frac{B I l}{4 K}$
(C) $\frac{B I l}{8 K}$
(D) $\frac{B I l}{16 K}$

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Key Concepts

Magnetic Force on a Current-Carrying Conductor

This concept explains that when a current flows through a conductor placed in a magnetic field, it experiences a force given by the Lorentz force equation F = I? x B, where I is the current, ? is the length of the conductor, and B is the magnetic field. The magnitude of this force depends on the sine of the angle between the current direction and the field, and its direction is determined by the right-hand rule.

Hooke's Law and Spring Mechanics

Hooke's Law states that the force exerted by a spring is directly proportional to its displacement from equilibrium, commonly expressed as F = kx, where k is the spring constant and x is the displacement. This concept is key to understanding how the spring extends or compresses in response to applied forces, providing the restoring force that balances other forces in mechanical systems.

Static Equilibrium in Mechanical Systems

Static equilibrium is a state in which the net force acting on a system is zero, resulting in no acceleration. In the context of the problem, it requires balancing magnetic forces, gravitational forces, and elastic (spring) forces. This equilibrium condition is essential for solving for the maximum elongation of the spring when the system is disturbed by an external force.

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