The current in the long, straight wire A B shown in Fig. 𝐄 29.7 is upward and is increasing stea

The current in the long, straight wire A B shown in Fig. 𝐄...

The current in the long, straight wire $A B$ shown in Fig. $\mathbf{E} 29.7$ is upward and is increasing steadily at a rate $d i / d t$. (a) At an instant when the current is $i,$ what are the magnitude and direction of the field $\vec{B}$ at a distance $r$ to the right of the wire? (b) What is the flux $d \Phi_{B}$ through the narrow, shaded strip? (c) What is the total flux through the loop? (d) What is the induced emf in the loop? (e) Evaluate the numerical value of the induced $\mathrm{emf} \quad$ if $\quad a=12.0 \mathrm{~cm}, b=36.0 \mathrm{~cm}$ $L=24.0 \mathrm{~cm},$ and $\mathrm{di} / \mathrm{dt}=9.60 \mathrm{~A} / \mathrm{s}$

Key Concepts

Integration in Electromagnetic Calculations

When the magnetic field varies over an extended area, calculating the total magnetic flux requires integrating the magnetic field over the surface. This integration sums infinitesimal contributions to yield the net flux, a necessary step in deriving induced emf for irregular or non-uniform fields.

Lenz's Law

Lenz's law determines the direction of the induced currents generated by a changing magnetic flux. It states that the induced emf and resulting current will flow in a direction that opposes the change in the magnetic flux that produced them, thereby conserving energy. This concept is key to predicting current directions in electromagnetic induction processes.

Magnetic Field of a Current?Carrying Wire

This concept uses Ampère's law to determine that the magnetic field (B) at a distance (r) from a long, straight current?carrying wire is given by B = ??I/(2?r). The direction of this field is given by the right-hand rule, meaning that with the thumb pointing in the direction of the current, the curled fingers show the field lines around the wire.

Magnetic Flux

Magnetic flux quantifies the total magnetic field passing through a given area, typically defined as the surface integral of the magnetic field over that area. It is a measure of how much of the magnetic field 'threads' through a surface and is fundamental in analyzing the interaction between magnetic fields and conductive loops.

Faraday's Law of Electromagnetic Induction

Faraday's law states that a time-varying magnetic flux through a closed loop induces an electromotive force (emf) in the loop. The induced emf is equal to the negative rate of change of the magnetic flux through the loop. This principle is central to understanding how varying magnetic fields can generate electrical currents.

Transcript

00:02 In the given problem here there is a straight current carrying wire which is carrying a current i a given instant and towards its right side there is a rectangular loop made up of a conducting wire whose length is given as l it's right it's right right arm is at a distance b from the straight conductor and its left arm is at a distance a from this straight conductor.

00:52 Then there is a small narrow strip, shaded strip, at a distance of r from the straight conductor.

01:11 Here the width of this small strip is dr.

01:16 The time rate of change of current through this conductor means rate of increase of current through this conductor is given as b i by d t.

01:39 Now in the first part of the problem we have to find an expression for the magnetic field at a distance are from this straight current carrying conductor and using biotrans.

01:53 And sever's law that magnetic field at our distance r is given as b is equal to mu not by 4 pi into 2 i by r or we can say b is equal to mu not by 2 pi into 1 by r.

02:36 In the second part of the problem, we have to find the small magnetic flux linked through this shaded strip.

02:50 So first of all, we will find area of the shaded strip and that is small area.

03:03 This is equal to length of the strip, which is equal to the length of the rectangular coil and its width into its width.

03:11 So this is l, d, r.

03:15 So magnetic flux linked through this strip, the small magnetic flux will be given by d5b.

03:37 As this is small, that's why we are taking it as the differentially small magnetic flux, means d5b.

03:45 And that will be given as the product of magnetic field at that point into area, with area.

03:53 Of the strip.

03:55 So here it becomes new knot by 2 pi into i by r into l d r.

04:05 So this is the answer for the second part of the problem.

04:10 The magnetic flux linked through this small strip, i l, d r.

04:13 In the third part of the problem, we have to find total magnetic flux linked through this small strip, i, l, d, r by r.

04:24 In the third part of the problem, we have to find total magnetic flux.

04:31 Linked through this rectangular coil for which we will integrate both the sides...

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