A conducting rod of length ℓ=35.0 cm is free to slide on...

A conducting rod of length $\ell=35.0 \mathrm{~cm}$ is free to slide on two parallel conducting bars, as shown in Figure P31.53. Two resistors $R_{1}=2.00 \Omega$ and $R_{2}=5.00 \Omega$ are connected across the ends of the bars to form a loop. A constant magnetic field $B=2.50 \mathrm{~T}$ is directed perpendicular into the page. An external agent pulls the rod to the left with a constant speed of $v=8.00 \mathrm{~m} / \mathrm{s}$. Find (a) the currents in both resistors, (b) the total power delivered to the resistance of the circuit, and (c) the magnitude of the applied force that is needed to move the rod with this constant velocity.

Key Concepts

Energy Conversion and Power Dissipation

When an emf is induced and a current flows through resistive components, electrical energy is converted into heat, manifesting as power dissipation in the circuit. Calculating the power delivered often involves using the relationship between voltage, current, and resistance. This concept highlights the interplay between mechanical work done in moving the conductor and the resulting electrical energy conversion into thermal energy.

Lorentz Force on Current-Carrying Conductors

The Lorentz force is the force exerted on a current-carrying conductor when it is placed in a magnetic field. This force is responsible for the mechanical interaction between the current in the rod and the magnetic field, and it is critical when considering the balance of forces required to maintain a constant velocity. The concept links the electrical and mechanical aspects of electromagnetic systems.

Ohm’s Law and Resistive Circuit Analysis

Ohm’s Law relates the voltage across a resistor to the current flowing through it and the resistance. In circuits where an induced emf appears, the current distribution can be determined by analyzing the resistive elements within the circuit. This analysis is essential for calculating the currents through various branches of a circuit with multiple resistors connected.

Faraday’s Law of Electromagnetic Induction

Faraday’s Law states that a changing magnetic flux through a closed circuit induces an electromotive force in the circuit. In the context of a moving conductor, the change in the area enclosed by the circuit or the motion relative to the magnetic field leads to a change in flux, which in turn induces an emf according to the law. This principle is central to the operation of many electrical devices and generators.

Motional Electromotive Force (emf)

This concept describes the emf generated when a conductor moves through a magnetic field. As the conductor cuts through magnetic field lines, charges inside it experience a Lorentz force, which causes a separation of charge and results in an induced voltage across the conductor. This is a foundational idea in understanding how mechanical motion can be converted into electrical energy.

Lenz’s Law

Lenz’s Law provides the direction of the induced current in such a way that it opposes the change in magnetic flux that produced it. This opposition ensures the conservation of energy and is the reason why the induced current creates its own magnetic field that counteracts the original change. It plays a crucial role in understanding the feedback mechanism in electromagnetic induction phenomena.

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