Rectangular Wire Loop A rectangular loop of wire with length...

Rectangular Wire Loop A rectangular loop of wire with length $a$, width $b$, and resistance $R$ is placed near an infinitely long wire carrying current $i$, as shown in Fig. $31-41 .$ The distance from the long wire to the center of the loop is $r$. Find (a) the magnitude of the magnetic flux encircled by the loop and (b) the amount of induced current in the loop $\left|i^{\text {ind }}\right|$ as it moves away from the long wire with velocity $\vec{v}$. (c) Indicate whether the induced current FIGURE $31-41$ w. is clockwise or counterclockwise. $\quad$ Problem $26 .$

Rectangular Wire Loop A rectangular loop of wire with length $a$, width $b$, and resistance $R$ is placed near an infinitely long wire carrying current $i$, as shown in Fig. $31-41 .$ The distance from the long wire to the center of the loop is $r$. Find (a) the magnitude of the magnetic flux encircled by the loop and
(b) the amount of induced current in the loop $\left|i^{\text {ind }}\right|$ as it moves away from the long wire with velocity $\vec{v}$. (c) Indicate whether the induced current FIGURE $31-41$ w. is clockwise or counterclockwise. $\quad$ Problem $26 .$

Key Concepts

Magnetic Flux

Magnetic flux is a measure of the total magnetic field passing through a given area. It is calculated by integrating the magnetic field over the surface area, taking into account the angle between the field lines and the normal to the surface. Understanding magnetic flux is essential for exploring how changes in the field or the area lead to induced effects in circuits.

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a circuit induces an electromotive force (emf) proportional to the rate of change of the flux. This law is a cornerstone of electromagnetism and explains how varying magnetic fields can generate electrical currents in conductive loops.

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 flow in a direction that opposes the change in flux. This principle is key for determining how systems respond dynamically to variations in their magnetic environment.

Magnetic Field of a Current-Carrying Conductor

The magnetic field around a current-carrying wire, such as an infinitely long straight conductor, is described by the Biot-Savart Law or Ampere's Law. This concept is essential for calculating the magnetic flux through a loop placed near the wire and forms the basis for understanding its influence on nearby conductive loops.

Induced Current and Circuit Resistance

When an emf is induced in a closed loop, the resulting current is determined by the loop's resistance according to Ohm's Law. This relationship is crucial for quantitatively predicting the behavior of circuits undergoing electromagnetic induction, including the magnitude and direction of the current flow.

Transcript

00:01 In the given problem there is a straight current carrying conductor.

00:10 This is the horizontal straight current carrying conductor which is carrying current towards left.

00:17 Let the current be high.

00:19 And then there is a rectangular loop made up of a conducting wire and this is the axis of this loop and the loop is oriented nearby this current carrying conductor in such a manner that its axis is at a distance of r from the straight current carrying conductor.

00:53 And then it is being taken away from this current carrying conductor with a velocity v.

01:00 Length of the loop rectangular loop is given as a small a here and its width is given as t first of all in the first part of the problem we have to find magnetic flux average magnetic flux due to this straight current connecting conductor which is linked through this rectangular loop so to find this magnetic flux linked through the coil we consider this rectangular loop to be made up of a large number of strips each having a differentially small width and then we choose one such strip differentially small strip having a width here suppose this is the differentially small strip this strip is at a distance of suppose it is its distance is x from the current carrying conductor and its width, differentially small width is x.

02:22 So length of the strip will be a only.

02:27 So differentially small area of the strip will be a.

02:43 D -a is equal to length into width.

02:46 Length is a and width we have assumed it to be d -a.

02:52 Now to find a magnetic flux linked through this strip, we will consider, we will find an expression for the magnetic field at the location of this strip which is at a distance of x from the current carrying conductor.

03:05 To magnetic field at the location of this small strip that is using by 10 severed law that magnetic field will be given by b is equal to mu not upon 4 pi to i by x the distance of the strip from the current gap connecting conductor.

03:42 Hence the small magnetic flux differentially small linked through this strip will be given by b5 is equal to b .a.

04:21 Here for a this will be da...

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