A rod of length ℓ, negligible resistance, and mass m slides...

A rod of length $\ell$, negligible resistance, and mass $m$ slides on two horizontal frictionless rails of negligible resistance by hanging a block of mass $m_{1}$ with the help of insulating a massless string passing through fixed massless pulley (as shown in Fig. 16.51). If a constant magnetic field $B$ acts upwards perpendicular to the plane of the figure, the terminal velocity of hanging mass is (A) $\frac{m_{1} g R}{B^{2} l^{2}}$ upward (B) $\frac{m_{1} g R}{B^{2} l^{2}}$ downward (C) $\frac{m_{1} g R}{2 B^{2} l^{2}}$ downward (D) $\frac{m_{1} g R}{B^{2} l}$ downward

A rod of length $\ell$, negligible resistance, and mass $m$ slides on two horizontal frictionless rails of negligible resistance by hanging a block of mass $m_{1}$ with the help of insulating a massless string passing through fixed massless pulley (as shown in Fig. 16.51). If a constant magnetic field $B$ acts upwards perpendicular to the plane of the figure, the terminal velocity of hanging mass is
(A) $\frac{m_{1} g R}{B^{2} l^{2}}$ upward
(B) $\frac{m_{1} g R}{B^{2} l^{2}}$ downward
(C) $\frac{m_{1} g R}{2 B^{2} l^{2}}$ downward
(D) $\frac{m_{1} g R}{B^{2} l}$ downward

Key Concepts

Electromagnetic Induction

This concept involves the generation of an electromotive force (emf) when a conductor moves through a magnetic field or when there is a change in magnetic flux through a circuit. Faraday's law quantifies this relationship, stating that the induced emf is proportional to the rate of change of magnetic flux. In systems with moving conductors, such as a sliding rod, the induced emf is a crucial factor that initiates currents which then interact with the magnetic field.

Lenz’s Law

Lenz’s law explains the direction of the induced current caused by a changing magnetic field, stating that the current will flow in such a way as to oppose the change in magnetic flux that produced it. This opposition is fundamental to electromagnetic damping, as the induced magnetic forces act to resist the motion that generated them, thereby affecting the dynamics of the system.

Lorentz Force

When a current-carrying conductor is placed in a magnetic field, it experiences a force known as the Lorentz force. This force is perpendicular to both the direction of the current and the magnetic field. In the context of a moving rod with induced currents, the Lorentz force provides the magnetic drag that opposes the motion, playing a key role in reaching a terminal velocity.

Terminal Velocity in Electromagnetic Damping

Terminal velocity is achieved when the net forces acting on a moving system balance, resulting in a constant speed. In electromagnetic systems, the gravitational force is balanced by the counteracting magnetic drag force generated by the induced currents. This dynamic equilibrium determines the steady-state or terminal velocity of the moving mass.

Resistive Circuit Dynamics

The presence of resistance in the conductive path influences the magnitude of the induced current, as dictated by Ohm's law. The circuit’s resistance, along with the induced emf from electromagnetic induction, determines how much current flows and consequently how strong the magnetic damping force is. This interplay between resistance and induced currents is critical for analyzing energy dissipation and force balance in such systems.

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