Two infinitely long solenoids (seen in cross-section) thread...

Key Concepts

Lenz's Law

Lenz’s law provides the direction of the induced current generated by a changing magnetic flux. It states that the induced current will flow in a direction that opposes the change in flux that produced it, ensuring the conservation of energy. This concept is essential for understanding the behavior and direction of induced currents in electromagnetic induction problems.

Ohm's Law

Ohm’s law relates the voltage (emf), current, and resistance in an electrical circuit. After determining the induced emf through Faraday’s law, Ohm’s law is used to compute the current by dividing the emf by the resistance in the circuit. It provides a direct connection between the electrical quantities in circuits with resistive components.

Properties of an Ideal Solenoid

An ideal solenoid is characterized by its ability to generate a uniform magnetic field inside its interior while having a negligible magnetic field outside. This simplification is useful in electromagnetic analyses because it allows the magnetic field within the solenoid to be treated as constant, thereby making the calculation of the magnetic flux straightforward.

Faraday's Law of Electromagnetic Induction

This principle states that a changing magnetic flux through a closed circuit induces an electromotive force (emf) in that circuit. The magnitude of the induced emf is proportional to the rate of change of the magnetic flux, and its direction is given by Lenz’s law. This concept is fundamental in analyzing situations where time-varying magnetic fields generate currents in circuits.

Magnetic Flux

Magnetic flux is the measure of the amount of magnetic field passing through a given area, and it is defined as the product of the magnetic field strength, the area it penetrates, and the cosine of the angle between the field and the area’s normal. In problems involving induction, calculating the change in magnetic flux is crucial for determining the induced emf.

Transcript

00:01 In the given problem, there are two loops which are connected with each other as shown here.

00:13 This is a resistance of 6 -on then there is a resistance of 3 -on and finally here.

00:28 This is one more resistance which is having a value of 5.

00:38 Here, in the first loop there is a small circular region.

00:47 Actually this is the cross section of a long infinite solenoid.

00:53 And in this solenoid, the magnetic field is shown with the help of crosses means the magnetic field is getting into the plane of paper.

01:03 And in the second loop, the magnetic field is being represented by the dots, which represent and the magnetic field is coming out of the plane of paper.

01:12 So due to this time varying magnetic field, as the flux will change through this square loop also, so a current will be induced as per faraday's laws of electromagnetic induction.

01:27 And using lens's law, the direction of current will be such that it will oppose the reasons behind its induction.

01:36 As a reason is this macro -magretic induction.

01:38 Which is inward and increasing so its opposition will be outward increasing magnetic field so in this whole region there should be an induced magnetic field coming out of it and as we know magnetic field comes out of the north pole so this square loop should behave like north pole means the direction of current induced in it should be counterclockwise now for the second the magnetic field which is changing through the solenoid is outward and it is increasing so its opposition should be inward increasing magnetic field inward means the magnet field should enter into it and as we know magnetic field enters into the south pole and south pole is given rise by a clockwise current so the current induced here in this loop should be clockwise.

02:43 So if we replace the induced emf with the battery, the direction of emf here in this loop will be like this and here it will be like this.

02:56 So first of all we should find the emfs induced in these loops which are being represented by e1 and e2.

03:09 Radius of this loop in the first square is r1 and in the second one it is r2 and these radii are given as r1 is 0 .10 meter and r2 is 0 .10 meter now first of all we will find the emf induced in the first loop which will be given by minus d phi by d t and if we look for the magnitude only then emf is equal to d by d t of b into a 1 area of this circular loop this solenoid cross section of this solenoid so it will become d by d t b into pi r1 square so so, pi r1 square can be taken as a constant out leaving behind d d b by d t so if i put these values 3 .14 0 .1 square and d b by d t has been given as 100 tesla per second so it comes out to be exactly 3 .14 volt now for the emf induced in the second loop e2 again the same formula we are going to use we are finding the magnitude so d by d t of b into a 2 so pi r2 square will come out d b by d t this is 3 .14 0 .15 square and into 100 test volt so then we calculate all these things it will come out to be 7 .065.

05:22 So this is the emf induced in the second loop.

05:30 Now we will change the circuit.

05:33 We will make an equivalent circuit diagram in which we will show the batteries as the sources of emf.

05:47 This is the resistance of 6 om.

05:51 This is the battery having emf e1.

05:55 Then this is the resistance of 3 -on.

06:02 Then this is the resistance of 5 -oam and this is second dmf induced, which is of 7 .065 volt.

06:18 E2 is equal to and e1 was 3 .14 volts.

06:27 So this is the equivalent circuit.

06:30 Now we will name it in order to apply kirchhoff's current law and voltage law.

06:36 This is a, b, c and d and e and f...