A cardboard tube is wrapped with windings of insulated wire...

A cardboard tube is wrapped with windings of insulated wire that are connected to a resistor. The tube is placed into a magnetic field that is changing with time. Determine whether the induced current in the resistor $R$ is directed from left to right or from right to left or is zero for (a) Figure 21.47 a with the magnetic field increasing, (b) Figure $21.47 \mathrm{~b}$ with the field decreasing, and (c) Figure $21.47 \mathrm{c}$ with the field increasing. (Consider only the magnetic flux through the tube.)

Key Concepts

Magnetic Flux

Magnetic flux is a measure of the total magnetic field passing through a given surface area. It depends on both the strength of the magnetic field and the orientation of the surface with respect to the field lines. Understanding magnetic flux is critical when applying Faraday's and Lenz's Laws, as changes in flux, whether due to intensity, area, or angle variations, are what lead to induced EMFs.

Electromagnetic Induction

This is the phenomenon by which a voltage or electromotive force (EMF) is generated in a conductor as a result of a change in the magnetic field surrounding it. When a magnetic field through a circuit changes with time, induced currents can be produced, and this principle underlies the workings of generators, transformers, and many other electromagnetic devices.

Faraday's Law

Faraday's Law quantifies electromagnetic induction, stating that the induced electromotive force in any closed circuit is equal to the negative of the rate of change of magnetic flux through the circuit. This law is fundamental in calculating the magnitude of the induced voltage when there is a time-dependent change in the magnetic field within a loop or coil.

Lenz's Law

Lenz's Law provides the direction of the induced current. It states that the induced current will flow in a direction such that its own magnetic field opposes the change in the original magnetic flux. This opposition is a manifestation of the conservation of energy and ensures that the induced effects counteract the flux changes that produced them.

Transcript

00:01 In the first part of the given problem, there is a solenoid found over a core.

00:18 This is the solenoid having a large number of turns like this.

00:33 This is the resistance joined with this solenoid.

00:38 A magnetic field is parallel to the solenoid and in a direction towards right.

00:56 The magnetic field is magnetic field which is towards right means left to right is increasing in magnitude.

01:23 So, the magnetic flux linked with the solenoid, each turn of the solenoid, will also increase due to which an emf and hence a current and hence a magnetic field will be induced.

01:48 Which direction as per lens's law should be opposite in nature opposing in nature to the inducing magnetic field and as the inducing magnetic field is from left to right and increasing so induced magnetic field should be from right to left means this induced magnetic field should be entering here to the right face means induced magnetic field should be entering to the right face of the coil of the solenoid.

03:56 So this right face should start behaving like south pole.

04:17 As we know, magnetity fill enters into the south pole.

04:21 So the direction of current in it, current induced in it, should be clockwise.

04:43 So if this current is clockwise, hence in the resistance it should be from left to right.

04:54 So in the resistance it should be from the first part of this.

05:17 Problem now in the second part of the problem the similar solenoid is there having the similar number of turns like this in the similar manner the similar resistance but the magnetic field this time is neither the parallel to solenoid nor or perpendicular to it is like this and it is decreasing so it will be be having two components.

05:53 One component along the length of the solenoid means it is from left to right and another component will be perpendicular to solenoid which will give rise to no magnetic flux linked through the solenoid...

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