The long, straight wire shown in Fig. P29.57a carries constant current I. A metal bar with lengt

step 1 Okay, so we're doing chapter 29, problem 57. Cool. So what we have here is we have the long infi

The long, straight wire shown in Fig. P29.57a carries...

The long, straight wire shown in $\textbf{Fig. P29.57a}$ carries constant current $I$. A metal bar with length $L$ is moving at constant velocity $\vec{v}$, as shown in the figure. Point a is a distance $d$ from the wire. (a) Calculate the emf induced in the bar. (b) Which point, $a$ or $b$, is at higher potential? (c) If the bar is replaced by a rectangular wire loop of resistance $R$ (Fig. P29.57b), what is the magnitude of the current induced in the loop?

Key Concepts

Motional EMF

Motional electromotive force refers to the voltage generated when a conductor moves through a magnetic field. This phenomenon occurs due to the Lorentz force acting on the free charges inside the conductor, which causes a separation of charge and thus develops an electric potential difference across the conductor. It is a foundational concept in electromagnetic induction and underlies many practical devices such as generators.

Magnetic Field of a Straight Current-Carrying Wire

A straight current-carrying wire produces a magnetic field that forms concentric circles around the wire. The magnitude of this magnetic field decreases with distance from the wire and can be determined using Ampère’s law. This concept is fundamental for analyzing the interaction between currents and magnetic fields in various electromagnetic scenarios.

Lorentz Force

The Lorentz force is the force exerted on a charged particle moving in a magnetic field. It acts perpendicular to both the velocity of the particle and the magnetic field, guiding the motion of charges and thereby generating phenomena such as motional EMF. This force is key to understanding how magnetic fields influence moving charges and current-carrying conductors.

Faraday's Law of Electromagnetic Induction

Faraday's law states that a change in magnetic flux through a circuit induces an electromotive force in the circuit, proportional to the rate of change of the flux. This principle is essential for explaining and calculating the induced emf in systems where the magnetic environment changes either by motion, variation in current, or a combination of both.

Ohm's Law

Ohm's law relates the voltage, current, and resistance in an electrical circuit through the formula V = IR. This relationship is crucial when analyzing circuits that incorporate induced emfs, as it allows the calculation of induced currents when the circuit’s resistance is known.

Lenz's Law

Lenz's law provides the direction of the induced current resulting from a change in magnetic flux, stating that the induced current will oppose the change causing it. This principle is essential in understanding the behavior of currents in electromagnetic induction processes, ensuring that energy conservation is maintained in the system.

Transcript

00:01 Okay, so we're doing chapter 29, problem 57.

00:05 Cool.

00:06 So what we have here is we have the long infinite wire up top with a current positive x direction here.

00:14 And a distance d away from it, we have a rod, the metal rod, that has length l, and that is a point a is the one closest, the end closest to the rod, and v is part of this away from the wire.

00:30 And we have, say, it's moving at a speed, v in the positive x direction.

00:35 Okay, so what we want to do now is find the emf induced on this.

00:43 So to do this, we know that emf is given us vbl, but b is now dependent on r, right? so we just have to, we're going to have to integrate this with r being pointing from the wire down the rod.

01:00 So this is our differential and if we're trying to find the total, we go, we go, we just have to integrate this.

01:04 This integrate this.

01:08 To do this, we'll have to figure out what b of r is.

01:18 And b of r and a wire we've used in some other problems that's given as u0 times the current through the wire all over 2 pi times the distance away.

01:29 So we can plug this in now.

01:32 And we see that this is going to be integral from d to d plus l because that's how we're starting.

01:40 The point a is when the wire starts.

01:42 So that's d away, and then b is where we end, which is d plus l away.

01:48 And now we just plug you on our stuff.

01:50 And this was mu not i, v over 2 pi, r, dr.

02:00 Cool...

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