A 0.650-m-long metal bar is pulled to the right at a steady...

A 0.650-m-long metal bar is pulled to the right at a steady 5.0 m/s perpendicular to a uniform, 0.750 T magnetic field. The bar rides on parallel metal rails connected through a 25.0-$\Omega$ resistor ($\textbf{Fig. E29.30}$), so the apparatus makes a complete circuit. Ignore the resistance of the bar and the rails. (a) Calculate the magnitude of the emf induced in the circuit. (b) Find the direction of the current induced in the circuit by using (i) the magnetic force on the charges in the moving bar; (ii) Faraday's law; (iii) Lenz's law. (c) Calculate the current through the resistor.

Key Concepts

Motional EMF

Motional EMF refers to the electromotive force generated when a conductor moves through a magnetic field. This phenomenon occurs due to the magnetic Lorentz force acting on the charge carriers within the conductor, which leads to a separation of charges and an induced voltage. The magnitude of the induced EMF is directly proportional to the velocity of the conductor, the strength of the magnetic field, and the length of the conductor within the field.

Faraday's Law

Faraday's Law of electromagnetic induction states that the induced electromotive force in a closed circuit is equal to the negative rate of change of the magnetic flux through the circuit. This law connects the concepts of magnetic flux, time-varying magnetic fields, and induced currents, and provides the fundamental principle behind how changing magnetic environments can induce electrical currents in conductors.

Magnetic Flux

Magnetic flux is a measure of the total magnetic field passing through a given area. It is dependent on the strength of the magnetic field and the orientation and size of the area through which the field lines pass. An understanding of magnetic flux is essential for applying Faraday's Law, as the induced EMF is directly related to the rate of change of magnetic flux through a circuit.

Lenz's Law

Lenz's Law provides the direction of the induced current resulting from electromagnetic induction. It states that the induced current will flow in such a direction that its own magnetic field opposes the change in the magnetic flux that produced it. This principle ensures the conservation of energy and is crucial for determining the polarity of the induced EMF in a circuit.

Magnetic Force on Moving Charges

The magnetic force on moving charges, described by the Lorentz force law, explains the force experienced by charge carriers moving within a magnetic field. This force is perpendicular to both the velocity of the charge and the magnetic field, and is the cause of charge separation in a moving conductor that leads to the creation of motional EMF. Understanding this concept is key to analyzing how currents are induced in conductors moving through magnetic fields.

Ohm's Law

Ohm's Law relates the current flowing through a conductor to the voltage across it and the resistance of the conductor. It provides a simple linear relationship where the current is equal to the voltage divided by the resistance. This law is essential for calculating the induced current in a circuit once the EMF and resistance are known.

Transcript

00:01 Here in this given problem there is a metal bar whose length that is given as 0 .650 meter.

00:23 It is moving with the uniform speed v is equal to 5 .0 meter per second perpendicular to a magnetic field 0 .750 tesla.

00:47 Resistance in the circuit that is 25 .0 ohm.

00:55 Here these are the two metal rails connected with each other with the resistance r whose value is 25 .0 ohm and this is the metal bar moving over the two rails with the velocity v.

01:17 And these are and this is the magnetic field uniform magnetic field directed into the plane of paper like this.

01:27 In the first part of the problem we have to find emf induced across the ends of this bar and that emf is given by the expression b v l product of magnetic field with the velocity and with the length of the metal bar.

01:52 For b this is 0 .750 multiplied by v which is 5 .0 multiplied by length which is 0 .650.

02:01 So, this emf is calculated to be equal to 2 .44 volt potential difference across the ends of the rod.

02:11 Answer for the first part of this given problem here.

02:15 In the second part of the problem we have to find direction of current induced and to find the direction of this current induced in the first method as the rod is moving rod and hence its free electrons are moving towards right.

03:15 So, electron will experience magnetic lorentz force in downward direction.

03:46 Downward means from end a of the metal bar to end b...

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