The magnetic flux through three different coils is changing as shown in Figure P21.17. For each

The magnetic flux through three different coils is changing...

Key Concepts

Graphical Interpretation of Induced emf

Graphing the induced emf involves understanding its relationship to the rate of change of magnetic flux over time. A steady magnetic flux produces zero induced emf, while rapid changes in flux result in higher emf values. Plotting these variations helps visualize how different flux change patterns affect the resulting induced emf, with the graph serving as a derivative representation of the flux variation.

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 such a direction that its own magnetic field opposes the change in the original magnetic flux, thereby preserving energy conservation within the system.

Electromagnetic Induction

Electromagnetic induction is the process by which a change in magnetic flux through a circuit induces an electromotive force (emf). This phenomenon is the basis for the operation of many electrical devices such as generators and transformers, and it demonstrates the fundamental link between electricity and magnetism.

Faraday's Law of Electromagnetic Induction

Faraday's Law quantifies the induced emf as proportional to the rate of change of magnetic flux through a circuit. Expressed mathematically as emf = -d?/dt, this law forms the cornerstone of understanding how and why an induced emf arises when the magnetic environment of a conductor changes over time.

Magnetic Flux

Magnetic flux is a measure of the quantity of magnetic field lines passing through an area. It is an essential concept in electromagnetism, calculated as the product of the magnetic field strength, the area, and the cosine of the angle between the field direction and the normal to the surface. This concept underlies the analysis of how flux changes can induce electrical effects in conductors.

Transcript

00:02 Hi, in the given problem in the first coil, variation of magnetic flux at a time has been given as this is the magnetic flux whose initial value was 0 .4 tesla meter square.

00:29 Then it reduces to 0 .2 tesla meter square over a time interval of 2 seconds and after which this flux becomes constant.

00:58 So corresponding to this variation of magnetic flux with the passage of time, we have to plot a variation of emf induced.

01:09 Again with the passage of time so using faraday's laws of electromagnetic induction which says emf induced is negative of t5 pi t t or we can say negative of phi 2 minus 5 1 final flux minus initial flux t 2 minus t 1 the range of the time so here for this time interval from 0 to 2 second it comes out to be negative of for 5 2 this is 0 .2 tesla into meter square minus 0 .4 tesla into meter square for the time this is 2 minus 0 so this is tesla into meter square for magnetic flux second for time hence is emf induced here comes out to be 0 .1 this is the value of this emf induced which is 0 .1 volt hence and this is positive so we can represent the graphical representation of this emf induced this is the emf induced after which the magnetic flux becomes constant so emf induced will be dropped to zero beyond and here this value is 0 .1 volt this is the curve between emf induced and the time and this is the answer for the first coil given in this problem now for the second coil the variation of magnetic flux is being given the time intervals are from 0 to 1 second first of all then it is from 1 to 2 second then 3 and then this is 4 second and the variation of magnetic flux is first of all it is increasing linearly for first 1 second then this magnetic flux becomes constant for the next 1 second and finally drops to 0 in next 2 seconds and this peak value here of this magnetic flux is 0 .4 tesla in 2 meter square.

04:12 And here this is the time which is measured in second.

04:16 So now we have to obtain a value, a variation of emf induced.

04:23 So using faraday's laws of electromagnetic induction, this is given as minus d -5 by dt.

04:33 Or we can say this is negative of final magnetic flux minus initial magnetic flux divided by the range of the time t2 minus t1 over which this change in magnetic flux has taken place so for this time interval from 0 to 1 second the time is 1 .1 minus 0 this is 1 second means and the magnetic flux is final magnetic flux is 0 .4 tisle meter square and initially it was zero tesla meter square divided by second so we can say the emf induced is having a constant value and it is constant and negative which comes out to be minus 0 .4 volt so if we draw the corresponding curve for the variation of emf induced with respect to time we will extend these time very variations over this curve also here.

05:47 So for the first time interval from 0 to 1 second, the emf induced is having a value of 0 .4 volt and this is negative.

05:59 So we can show it here.

06:05 This is 0 .4 volt and negative.

06:10 So this is the graphical representation for the magnetic for the magnetic for the emf induced now for the time interval of from one second to two second as the magnetic flux linked to the coil becomes constant.

06:30 So emf induced now will become zero and beyond which now the flux will drop.

06:37 So the emf induced now will be again negative of 5 -2 minus 5 -1 for 5 -2 this time this is 0 and for 5 -1 this is 0 .4 and for time this is 2 second this time.

06:51 Tesla meter squared per second.

06:54 So finally, this emf induced now over the time interval of two second to four second comes out to be positive 0 .2 volt means half of this 0 .4 volt.

07:08 If you can mark it here, then the graphical representation of this emf induced is this here, which is having a value of 0 .3...

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