The conducting rod ab shown in Fig. 298 makes contact with...

The conducting rod ab shown in Fig. 298 makes contact with metal rails $c a$ and $d b$ . The apparatus is in a uniform magnetic field of 0.800 T, perpendicular to the plane of the figure (a) Find the magnitude of the emf induced in the rod when it is moving toward the right with a speed 7.50 $\mathrm{m} / \mathrm{s}$ . (b) In what direction does the current flow in the rod? (c) If the resistance of the circuit $a b d c$ is 1.50$\Omega$ (assumed to be constant), find the force (magnitude and direction) required to keep the rod moving to the right with a constant speed of 7.50 $\mathrm{m} / \mathrm{s}$ . You can ignore friction. (d) Compare the rate at which mechanical work is done by the force $(F v)$ with the rate at which thermal cnergy is developed in the circuit $\left(I^{2} R\right) .$

Key Concepts

Lenz's Law

Lenz's Law states that the induced current in a conductor will flow in such a direction that its magnetic field opposes the change in magnetic flux that produced it. This is a manifestation of the conservation of energy and helps determine the direction of the induced current in electromagnetic induction problems.

Joule Heating and Energy Conversion

Joule heating describes the process by which electrical energy is dissipated as heat in a resistor due to the current flow. In electromechanical systems, the mechanical work done by an external force to maintain motion against magnetic forces is converted into electrical energy, which in turn is turned into thermal energy in the resistor. Comparing the mechanical power input (force times velocity) with the electrical power dissipated (I^2R) illustrates energy conservation within the system.

Lorentz Force

The Lorentz Force is the force exerted on a charged particle moving in a magnetic field, which is given by the cross product of the particle's velocity and the magnetic field. In a current-carrying rod, this force can act to slow the motion or require an external force to maintain a constant speed, directly linking the mechanical and electromagnetic aspects of the system.

Faraday's Law of Electromagnetic Induction

Faraday's Law explains how a change in magnetic flux through a circuit induces an electromotive force. In the context of a moving conductor, the movement changes the magnetic flux linking the circuit over time, thereby inducing an emf. This law provides the quantitative relationship necessary to calculate the induced voltage in scenarios involving motion.

Motional Electromotive Force (emf)

This concept refers to the voltage induced in a conductor when it moves through a magnetic field. As the rod moves, its charge carriers experience a magnetic force that separates positive and negative charges, creating a potential difference along the rod. The magnitude of this emf depends on the speed of the rod, the magnetic field's strength, and the effective length of the conductor within the magnetic field.

Transcript

00:01 So firstly for this problem, we'll calculate the magnitude of the induced emf across the ends of the rod.

00:09 So epsilon, the adduced emf, is equal to v, the speed of the rod, times the strength of the magnetic field b, times the length of the rod l.

00:18 And these values are known.

00:21 So this is 7 .5 meters per second, or v, the magnetic field strength is given as 0 .8.

00:31 Tesla's and the length of the rod is 0 .5 meters.

00:38 And hence we get an induced emf of three balls.

00:48 Next we want to find the direction of the induced current.

00:54 Well we know that the magnetic field b is into the page.

01:00 So that's its direction.

01:01 The flux increases as the bar moves to the right.

01:04 So the magnetic field of the induced current is out of the page inside the circuit.

01:11 So the induced magnetic field will call it the induced or the magnetic field due to the induced current is out of the page.

01:22 To produce the magnetic field in this direction, the induced current must be in a counterclockwise direction.

01:33 And so that's from b to a in the rod.

01:48 Next we want to calculate the force that needs to be applied to provide this motion.

01:55 We know that the current i is equal to the induced emf epsilon divided by the resistance.

02:03 Again, both these values are known...

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