A bar of mass m, length d, and resistance R slides without...

A bar of mass $m$, length $d$, and resistance $R$ slides without friction on parallel rails, as shown in Figure P31.55. A battery that maintains a constant $\operatorname{emf} \boldsymbol{\varepsilon}$ is connected between the rails, and a constant magnetic field $\mathbf{B}$ is directed perpendicular to the plane of the page. If the bar starts from rest, show that at time $t$ it moves with a speed $$v=\frac{\boldsymbol{\varepsilon}}{B d}\left(1-e^{-B^{2} d^{2} t / m R}\right).$$

A bar of mass $m$, length $d$, and resistance $R$ slides without friction on parallel rails, as shown in Figure P31.55. A battery that maintains a constant $\operatorname{emf} \boldsymbol{\varepsilon}$ is connected between the rails, and a constant magnetic field $\mathbf{B}$ is directed perpendicular to the plane of the page. If the bar starts from rest, show that at time $t$ it moves with a speed
$$v=\frac{\boldsymbol{\varepsilon}}{B d}\left(1-e^{-B^{2} d^{2} t / m R}\right).$$

Key Concepts

Electrical Resistance in Circuits

Electrical resistance is a measure of how strongly a material opposes the flow of electric current. In circuits, resistors dissipate energy in the form of heat and play a critical role in determining the current distribution and the voltage drops across various components. Resistance, together with sources of emf and other circuit elements, is crucial in modeling the temporal evolution of the current and, by extension, impacts the electromagnetic forces acting on moving conductors.

First-Order Differential Equations

First-order differential equations often arise in the analysis of circuits and mechanical systems where the rate of change of a quantity, like velocity or current, is proportional to the quantity itself. These equations typically yield solutions that involve exponential functions, reflecting processes such as the charging of a capacitor, the discharge of a battery, or the damping of motion. Such equations are essential for predicting the transient response of systems before they reach a steady state.

Newton's Second Law in Electromechanical Systems

Newton's Second Law, which relates force, mass, and acceleration, is fundamental in analyzing the dynamics of electromechanical systems. In problems involving moving conductive elements under magnetic influence, the mechanical forces derived from electromagnetic effects (such as those from the Lorentz force) are balanced with inertial and frictional forces to determine the motion of the object. This law helps bridge the gap between electrical phenomena and mechanical motion, leading to differential equations that describe system behavior over time.

Lenz's Law

Lenz's Law is a principle that states the direction of an induced current is such that it creates a magnetic field opposite to the change in magnetic flux that produced it. This law is essentially a statement of the conservation of energy and serves to determine the polarity of the induced emf. Lenz's Law explains the self-regulating nature of electromagnetic systems, where the induced effects tend to oppose the motion or change causing them, thereby acting as a natural damping mechanism.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in an electromagnetic field. In the context of current-carrying conductors or moving charges, it is the combination of electric and magnetic forces that acts on the particles. This force not only determines the direction and magnitude of particle motion but also plays a crucial role in the working of devices like motors and generators, where mechanical movement is produced or resisted by electromagnetic interactions.

Motional EMF

Motional emf is a specific form of electromagnetic induction that occurs when a conductor moves through a magnetic field. The movement causes free charges within the conductor to experience a Lorentz force, which separates positive and negative charges, thus creating a potential difference across the conductor. This concept is frequently utilized in applications such as electric generators and is essential for understanding the interplay between mechanical motion and electrical energy generation.

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic environment within a closed loop induces an electromotive force (emf) or voltage. This fundamental principle, which underpins transformers, generators, and various sensing devices, is governed by Faraday's law. It highlights that a relative motion between a conductor and a magnetic field, or a temporal change in the field itself, can produce an induced current in the conductor.

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