The conducting rod ab shown in Fig. E29.27 makes contact...

The conducting rod ab shown in Fig. E29.27 makes contact with metal rails $c a$ and $d b .$ The apparatus is in a uniform magnetic field of 0.800 $\mathrm{T}$ , 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 energy is developed in the circuit $\left(I^{2} R\right)$ .

Key Concepts

Energy Conversion and Mechanical Work in Electromagnetic Systems

In systems involving electromagnetic induction, the work done by external forces to maintain constant motion against the electromagnetic damping (often provided by the induced current) is converted into electrical energy that is subsequently dissipated as thermal energy (I²R). This energy conversion principle, which equates mechanical power (force times velocity) to the rate of electrical energy dissipation, ensures the conservation of energy within the system.

Lenz's Law

Lenz's Law provides the direction of the induced current resulting from electromagnetic induction. It states that the induced current creates a magnetic field that opposes the change in magnetic flux causing the induction. This concept is key for understanding and predicting the directional behavior of induced currents in moving conductors.

Ohm's Law

Ohm's Law, expressed as V = I * R, relates the voltage across a conductor to the current flowing through it and its electrical resistance. It is fundamental for calculating the current induced in a circuit when a motional emf is applied, and for understanding energy dissipation as heat due to resistive elements.

Motional Electromotive Force (emf)

When a conductor moves through a magnetic field, the free charges inside the conductor experience a force that separates them along its length, generating a potential difference. This phenomenon, known as motional emf, is typically calculated as ? = B * L * v, where B is the magnetic field strength, L is the length of the conductor within the field, and v is its velocity across the field.

Lorentz Force

Charged particles moving in a magnetic field experience the Lorentz force, given by F = q(v × B), which is responsible for deflecting the charges in the conductor. This deflection leads to the accumulation of charge at the ends of the rod, thereby establishing the induced emf and determining the direction of current flow.

Transcript

00:01 Hi, here in this given problem, these are the two metal rails ac and bd.

00:14 And over these two metal rails, the single rod ab that is allowed to move towards right with the velocity v.

00:29 The system is within a uniform magnetic field directed into the plane of paper like this.

00:43 In the first part of the problem, magnitude of this magnetic field that is 0 .800 tesla, length of the rod between the two rails, this is 50 .0 centimetre or we can say this is 0 .50 metre in mk system.

01:04 V, the speed with which the rod is moving towards right that is given as 7 .50 metre per second.

01:14 So, in the first part of the problem, motional emf induced across this rod ab that will be given by the expression v multiplied by v multiplied by l.

01:44 Means this is 0 .800 tesla multiplied by v 7 .50 multiplied by length 0 .50.

01:54 So, it is calculated to be equal to 3 .0 volt, answer for the first part of this given problem.

02:04 Now in the second part of the problem, we have to calculate the direction of the current induced in this rod.

02:20 So, using fleming's right hand rule, direction of this current in the rod that is up along b to a.

03:03 Then in the third part of the problem, this time resistance of whole of the circuit that is given as 1 .50 ohm.

03:11 So, current will be given by using ohm's law, emf induced divided by r.

03:19 So, the force required to keep the rod moving that will be given by an expression for magnetic force bil.

03:30 Means this is b into e by r multiplied by l...

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