Rod on Rails A metal rod is forced to move with constant velocity v⃗ along two parallel metal ra

Rod on Rails A metal rod is forced to move with constant...

Rod on Rails A metal rod is forced to move with constant velocity $\vec{v}$ along two parallel metal rails, connected with a strip of metal at one end, as shown in Fig. $31-42 . \mathrm{A}$ magnetic field of magnitude $|\vec{B}|=$ $0.350 \mathrm{~T}$ points out of the page. (a) If the rails are separated by $25.0 \mathrm{~cm}$ and the speed of the rod is $55.0$ $\mathrm{cm} / \mathrm{s}$, what emf is generated? (b) If the rod has a resistance of $18.0$ $\Omega$ and the rails and connector have negligible resistance, what is the current in the rod? (c) At what rate is mechanical energy being transformed to thermal energy?

Key Concepts

Joule Heating (Resistive Dissipation)

Joule heating refers to the conversion of electrical energy into thermal energy when current flows through a resistive material. The rate at which this energy is dissipated as heat can be calculated using the formula P = I²R or P = emf × I, emphasizing how induced currents lead to mechanical energy (input) converting into thermal energy in a resistive circuit.

Ohm's Law

Ohm's Law is a fundamental principle in electrical circuits that relates the current flowing through a conductor to the voltage across it and the conductor's resistance, expressed as V = IR. This relation is crucial for computing the current produced by the induced emf when the resistance of the circuit is known.

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a circuit induces an electromotive force (emf) in the circuit. In the context of a moving rod, even though the magnetic field itself is constant, the motion of the rod causes a change in the area enclosed by the circuit, effectively changing the magnetic flux and thus inducing an emf.

Motional Electromotive Force (emf)

This concept refers to the voltage generated when a conductor moves through a magnetic field. Due to the motion, free charges within the conductor experience a magnetic force, leading to charge separation and hence an induced voltage. The magnitude of this motional emf is determined by the magnetic field strength, the effective length of the conductor in the field, and the component of velocity perpendicular to the field.

Transcript

00:01 In the given problem there are two parallel metallic rails which have been joined here at one end of the rails and there is a metallic rod which is forced to move towards left with the velocity v.

00:28 The gap between the rails which will serve as the length of this rod that is given as l and there is a magnetic field in this setup which is directed perpendicular to the plane of paper and coming out of it here the magnitude of this magnetic field is given as 0 .350 tesla and that is out of the plane of paper length of the rod metallic rod that is centimeter which may also be written as 0 .25 meter.

01:13 It is moving to the left with the velocity 55 .0 centimeter per second which may also be written as 0 .5 meter per second now in the first part of the problem we have to find emf induced across the ends of this metallic rod and that will be known as motionally mf induced and that motional emf induced across the ends of metallic rod that is given as e is equal to the product of magnetic field in which it is moving with its velocity and with the length the rod so plugging in all known values here this emf induced will be given by 0.

02:18 350 tesla for the magnetic field 0 .5 meter per second for the velocity of this metallic rod and for the length of the rod 0 .25 meter.

02:32 So finally this cmf induced here comes out to be 0 .0481 volt or we can say this is 48 .1 millivolt which is answer for the first part of this problem.

02:55 Now in the second part of the problem, we have to find current passing through this loop, closed loop, or through this rod.

03:06 For which resistance of the rod is given as r is equal to 18 .0 oom.

03:15 So using ooms law, we get an expression...

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