Two closed loops A and C are close to a long wire carrying a...

Two closed loops $A$ and $C$ are close to a long wire carrying a current $I$ (Fig. $\mathbf{E 2 9 . 1 7}$ ). (a) Find the direction (clockwise or counterclockwise) of the current induced in each loop if $I$ is steadily decreasing. (b) While $I$ is decreasing, what is the direction of the net force that the wire exerts on each loop? Explain how you obtain your answer.

Two closed loops $A$ and $C$ are close to a long wire carrying a current $I$ (Fig. $\mathbf{E 2 9 . 1 7}$ ).
(a) Find the direction (clockwise or counterclockwise) of the current induced in each loop if $I$ is steadily decreasing. (b) While $I$ is decreasing, what is the direction of the net force that the wire exerts on each loop? Explain how you obtain your answer.

Key Concepts

Magnetic Force on Current Loops

When a current-carrying loop is placed in an external magnetic field, it experiences forces and torques due to the interaction between the current in the loop and the magnetic field. In cases where the field is non-uniform, these forces can result in a net movement of the loop, which can be analyzed by considering the magnetic dipole moment of the loop and how it interacts with the gradient of the external magnetic field.

Magnetic Fields from Current-Carrying Conductors

A current-carrying conductor, such as a long wire, produces a magnetic field that encircles the wire according to the right-hand rule. Understanding the geometry and direction of this field is crucial when analyzing how nearby circuits or loops interact with the field, both in terms of induced currents and magnetic forces.

Lenz's Law

Lenz's Law provides the direction of the induced current by stating that the induced current will flow in such a way that its own magnetic field opposes the change in the magnetic flux that produced it. This opposition is a manifestation of the conservation of energy, ensuring that the induced effects resist the change in the original magnetic environment.

Faraday's Law

Faraday's Law of Induction explains that a changing magnetic flux through a closed circuit induces an electromotive force (EMF) in the circuit. This principle is essential to understanding how any time variation in the magnetic field around a loop gives rise to an induced current, the magnitude of which is proportional to the rate of change of the magnetic flux through the loop.

Magnetic Flux

Magnetic flux is a measure of the total magnetic field passing through a given area. In the context of induction problems, the change in magnetic flux through a loop (due to either a changing external field or changing area/orientation of the loop) is the driving factor behind the induced electromotive force, which in turn determines the behavior of the induced current.

Transcript

00:01 In the given problem, first of all here there is a straight conductor, current carrying conductor, carrying a current i at an instant.

00:15 And there are two loops in the neighborhood of this current carrying conductor.

00:20 The first loop named as c is above this straight conductor and another loop named as a, that is below.

00:32 This is straight conductor.

00:34 Here there is here this is c and this is a.

00:38 Now there will be a magnetic field all around this is straight current carrying conductor as per barotent sever's law, whose direction using right hand thumb rule will be outward perpendicular to the plane of paper in the region above the conductor means the magnetic field which is linked through this square loop the tangular loop that is coming out of the plane of paper.

01:11 And similarly, using the same rule, right -hand thumb rule, the direction of magnetic field will be into the plane of paper means into this loop a, the magnetic field is entering.

01:25 And as the current i is reducing with the passage of time.

01:41 So the magnetic field around the conductor will also be decreasing with the time.

02:10 Means to say the magnetic flux linked through both the loops a and c will be decreasing.

02:42 So as per paradise laws of electromagnetic induction, the currents will be induced in both the loops which will create their own magnetic field whose directions as per level, as per lenses law should be opposite to the magnetic field due to which these are getting induced.

04:15 For the loop a as the reason behind the current induced is inward decreasing magnetic field.

04:54 Its opposition should be inward increasing magnetic field...

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