A metal bar with length L, mass m, and resistance R is...

A metal bar with length $L,$ mass $m,$ and resistance $R$ is placed on friction less metal rails that are inclined at an angle $\phi$ above the horizontal. The rails have negligible resistance. A uniform magnetic field of magnitude $B$ is directed downward as shown in Fig. $\mathbf{P 2 9 . 6 9 .}$ The bar is released from rest and slides down the rails. (a) Is the direction of the current induced in the bar from $a$ to $b$ or from $b$ to $a ?$ (b) What is the terminal speed of the bar? (c) What is the induced current in the bar when the terminal speed has been reached? (d) After the terminal speed has been reached, at what rate is electrical energy being converted to thermal energy in the resistance of the bar? (e) After the terminal speed has been reached, at what rate is work being done on the bar by gravity? Compare your answer to that in part (d).

A metal bar with length $L,$ mass $m,$ and resistance $R$ is placed on friction less metal rails that are inclined at an angle $\phi$ above the horizontal. The rails have negligible resistance. A uniform magnetic field of magnitude $B$ is directed downward as shown in Fig. $\mathbf{P 2 9 . 6 9 .}$ The bar is released from rest and slides down the rails.
(a) Is the direction of the current induced in the bar from $a$ to $b$ or from $b$ to $a ?$ (b) What is the terminal speed of the bar? (c) What is the induced current in the bar when the terminal speed has been reached? (d) After the terminal speed has been reached, at what rate is electrical energy being converted to thermal energy in the resistance of the bar? (e) After the terminal speed has been reached, at what rate is work being done on the bar by gravity? Compare your answer to that in part (d).

Key Concepts

Electromagnetic Induction

Electromagnetic induction refers to the generation of an electromotive force (emf) in a conductor when it experiences a change in magnetic flux. This phenomenon underpins the behavior of circuits involving moving conductors in magnetic fields. In the context of the problem, as the metal bar moves along the rails, the area of the loop—and hence the magnetic flux through it—changes, which induces an emf and a current in the bar.

Motional EMF

Motional emf is the specific type of electromotive force that arises when a conductor moves through a magnetic field, leading to charge separation along the conductor. The magnitude of this emf is typically given by the product of the magnetic field strength, the length of the conductor, and its velocity across the magnetic field. It is a key concept in calculating the induced voltage in the moving bar of the problem.

Lenz's Law

Lenz's Law states that the direction of the induced current is such that its own magnetic field opposes the change in magnetic flux that produced it. This gives rise to a resistive or braking force on the moving conductor, ensuring energy conservation within the system. In the described problem, Lenz's Law helps determine the direction of the induced current in the metal bar as it slides down the rails.

Terminal Velocity in Electromagnetic Systems

Terminal velocity in electromagnetic systems occurs when the force driving the motion, in this case gravity, is exactly balanced by opposing forces, such as the magnetic drag resulting from the induced current (or the back emf). At terminal velocity, there is no net acceleration. This balance is central to parts of the problem that ask for the terminal speed of the bar, where gravitational work is converted into electromagnetic and eventually thermal energy.

Energy Conversion and Dissipation in Resistive Circuits

When the bar reaches terminal velocity, the mechanical energy provided by gravity is continuously converted into electrical energy via electromagnetic induction and then dissipated as thermal energy in the resistance of the bar. The rate of energy conversion and subsequent dissipation can be analyzed by comparing the power input from gravity with the electrical power dissipated (calculated using the square of the current multiplied by the resistance), ensuring that energy is conserved in the system.

Transcript

00:01 In the given problem, these are the two rails, parallel rails, which are inclined at an angle of phi with the horizontal.

00:17 So this is the angle of their inclination with the horizontal, with the horizontal, which is phi.

00:26 Then there is a metallic bar sliding over these two rails in downward direction.

00:40 Then there is a magnetic field which is acting in a vertically downward direction like this everywhere within this setup of the two rails.

00:58 So these are the magnetic field lines.

01:01 Now it is given that the length of metal bar is given as l, its mass is m, the resistance is r, and this downward magnetic field is having a magnitude of b.

01:37 Now, in the first part of the problem, we have to find when this metal bar will be sliding down, the current induced in which direction there will be the current induced into this metal metallic bar.

01:55 So first of all, when this bar will be sliding down here like this, first of all, there is the weight of this bar acting vertically downward.

02:11 So its component perpendicular to the rails will be having a component mg kos 5 and another component along the rails will be m .g.

02:26 Sign 5 acting downward along the rails.

02:31 So when this rod will be moving down under the influence of this component, mg sine 5, this rod will be, this metallic bar will be intercepting the magnetic field lines.

02:44 But first of all, we will find the component of magnetic field lines also which will be intercepted by the metal bar and that component will be this here.

03:00 If this angle is 5, this angle will also be 5.

03:04 So the component of magnetic field will be b cos 5.

03:10 Now the bar will be moving perpendicular to this component b cos 5.

03:16 Hence if we use clemmings right hand rule here.

03:32 The direction of current induced in the metal bar will be from a to b.

03:54 And it becomes the answer for the first part of the problem.

03:59 Means there will be a current in this bar in a direction from a to be let this current be i now when this bar will be moving downward and it will be having a current i as shown here then this current carrying bar will experience a magnetic force also in part b this current carrying metal bar will experience a magnetic force whose direction as per fleming's left -hand rule will be rightward.

05:18 Here, as shown in the figure, the magnetic force experienced by this bar will be like this.

05:29 F, which will be given by the expression b -i -l, where l is the length of this metal bar.

05:36 So it is clear that this force will be at an angle 5 with the rails.

05:46 So this force will be having a component f cos 5 upward along the rails.

05:56 Now in case of terminal speed at terminal speed there will be no acceleration in the bar means no net force means the downward force acting on the bar will be equal to upward force.

06:31 And we have shown both of these forces in the figure.

06:34 Downward force is m .g.

06:36 Sine phi and upward forces f cos phi means bil cos phi.

06:42 Hence we can say m .g sine phi is equal to b i .l into cost.

06:52 5.

06:57 Now for current using om's law that will be given by the ratio of emf induced, motional emf induced with its resistance into l cos phi...

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