The long, straight wire shown in Fig. P29.55a carries...

The long, straight wire shown in Fig. P29.55a carries constant current $I$. A metal bar with length $L$ is moving at constant velocity $\overrightarrow{\boldsymbol{v}},$ 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. $\mathrm{P} 29.55 \mathrm{~b}$ ), what is the magnitude of the current induced in the loop?

Key Concepts

Ohm's Law in Inductive Circuits

Ohm's law relates the induced EMF in a circuit to the current and resistance in that circuit. In inductive scenarios, once an emf is induced (by motion or change in flux), the resulting current is determined by the circuit's overall resistance, allowing for quantification of induced currents.

Lenz's Law

Lenz's law provides the direction of the induced current resulting from a changing magnetic flux, asserting that the induced current will always flow in such a way as to oppose the change in flux that produced it. This principle is crucial for correctly determining potential differences in circuits with induced emf.

Faraday's Law of Electromagnetic Induction

Faraday's law states that an electromotive force is induced in a circuit when there is a change in the magnetic flux linking the circuit. In the context of moving conductors, the relative motion between the conductor and the magnetic field leads to a change in flux, which in turn induces an EMF.

Motional Electromotive Force (EMF)

Motional EMF refers to the voltage induced in a conductor moving through a magnetic field. This phenomenon is explained by the Lorentz force acting on the free charge carriers in the conductor, which separates charges along its length and creates a potential difference across its ends.

Magnetic Field of a Long Straight Conductor

This concept describes how a steady current in a long, straight wire produces a magnetic field in the surrounding space. The field strength typically decreases with distance from the wire and its direction is given by the right?hand rule. This field is fundamental when analyzing interactions with moving conductors placed near the wire.

Transcript

00:01 In the given problem here, this is the straight current carrying conductor below which there is a metal bar, put like this, perpendicular to the length of this current carrying conductor.

00:25 The current carried by this conductor is given as i.

00:28 The upper end of this steel bar is at a distance d from the wire.

00:39 Length of this metal bar is given as l and it is moving towards right with the velocity v.

00:51 Then there is a small segment of this metal bar taken at a distance.

01:01 Distance are from the current carrying conductor.

01:06 The width of this small element is dr.

01:12 So magnetic field at a distance are from the straight current carrying conductor is given as using by 10th severed law that is given as mu not by 2 pi into i by r.

01:54 So within this magnetic field, then this small segment having the width, dr moves towards right with the velocity v, then motional emf induced across the width of the small element will be given by let it be b e, emotional emf, and that is given as b into v into br, width of that small element.

02:46 So it becomes mu not by 2 pi into i by r into v into d r.

03:01 So the net emf induced across this metal bar will be given by integrating both the sides.

03:23 So that net emf induced will be given as integration of this mu not by 2 pi i by r, v, dr, having the limits from d to d plus l, taking the constants out which include mu not by 2 pi into i into v.

03:52 Leaving behind the integration of 1 by r with respect to dr having the limits from d to d plus l.

04:01 So this integration comes out to be mu not by 2 by 2...

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