The conducting rod a b shown in Fig. E29.27 makes contact...

The conducting rod $a b$ 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 Conservation in Electromagnetic Systems

This concept involves the conversion of mechanical work into electrical energy and thermal energy within a system exhibiting electromagnetic induction. When a force does work on a moving conductor in a magnetic field, the mechanical energy is partly converted into electrical energy (manifested as an induced EMF and current) and dissipated as heat due to resistive losses in the circuit. The balance of these energies is crucial in understanding the efficiency and performance of electromagnetic devices.

Lenz's Law

Lenz's Law provides the direction of the induced current resulting from an EMF, stating that the current will flow in such a direction as to oppose the change in magnetic flux that produced it. It is a manifestation of the conservation of energy principle, ensuring that the induced magnetic effects counteract the motion or change that produced them, thereby resisting the alteration in the system.

Magnetic Force on a Current-Carrying Conductor

When a current-carrying conductor is placed in a magnetic field, it experiences a force perpendicular to both the direction of the current and the magnetic field. This is described by the Lorentz force law. In an induction setup, this force can oppose the motion of the conductor, necessitating an external force to maintain a constant speed, and this interaction is fundamental in devices that convert mechanical work into electrical energy (or vice versa).

Faraday's Law of Induction

Faraday’s Law states that an electromotive force is induced in a closed circuit when there is a change in the magnetic flux through the circuit. In the context of a moving conductor, the change in flux can arise from the conductor’s motion in a uniform magnetic field, leading to the establishment of an induced EMF. This law is foundational in understanding how mechanical motion can create electrical energy.

Motional Electromotive Force (EMF)

This concept describes the voltage generated in a conductor moving through a magnetic field. The conductor, cutting through magnetic field lines, experiences an induced EMF that is proportional to its speed, the magnetic field strength, and its effective length within the field. This phenomenon is a direct application of electromagnetic induction where motion plays the key role in generating electrical potential differences.

Transcript

00:01 Hi in the given problem here this is a uniform magnetic field directed perpendicularly into the plane of paper.

00:16 Within this magnetic field there are two straight rails, metallic rails, ca and db over which this straight conductor is moving to the velocity v, right word.

00:37 This is d, b, a, and c.

00:43 The magnitude of this magnetic field is given as 0 .8 -00 tesla.

00:51 Length of the rod will be equal to the gap between the two metallic rails, which is given as 50 .0 centimeter.

01:00 So length of this rod will be 50 .0 centimeter or we can say this is 0 .50 meter.

01:11 The velocity of this rod is 7 .50 meter per second.

01:19 Now in the first part of the problem, we have to find the emotional emf induced across the ends of the rod, which is given as the product of magnetic field with the velocity and with the length of the conductor.

01:32 The effective length of the conductor which is moving perpendicular to the magnetic field.

01:38 So plugging in on known values, this is 0 .80 -t tesla multiplied by 7 .50 meter per second multiplied by length, which is 0 .50 meter.

01:54 So finally, this emf induced across the ends of the conductor comes out to be 3 .00 volt and it becomes the answer for the first part of the problem.

02:07 Now in the second part of the problem we have to find the direction of current induced.

02:15 So using lemmings right hand rule the direction of current induced in this raw comes out to be from b to a.

02:59 So this is the answer for the second part of the problem.

03:02 Now in the third part of the problem, resistance of whole of this loop, rectangular loop, is given as 1 .50 on.

03:13 So magnetic force experienced by this rod will be given by the product of magnetic field with the current passing through the loop.

03:27 Loop and the length of this rod...

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