Faraday's Law

Transcript

0:00Introduction

Hello everyone! Let's start with a small puzzle. Take a closed wire loop sitting in a magnetic field. Pull it out of the field slowly, and some charge flows around it. Now pull it out twice as fast. Does twice the charge flow? Most students say yes, straight away. Hold on to your answer; by the end of this video, you will know exactly what doubles and what does not. This is Faraday's law, and it comes up often in JEE, so let's begin.

0:32Faraday's law: a changing flux induces an emf

So, let me ask you something first. You hold a strong magnet perfectly still inside a coil. Is any emf induced in it? No, there is no emf at all! Faraday's law says: whenever the magnetic flux linked with a circuit changes, an emf is induced in it. The double arrow means which gives, and the key word is changes. Okay? So how big is this emf? The curly E is the emf, and capital phi, the Greek letter, is the magnetic flux through one turn, in weber. Curly E equals minus N times d phi over d t, the rate at which the flux changes. Right? Here, N is the number of turns, and N times phi is called the flux linkage. With me? Now look at that minus sign in front. What do you think it is telling us? It gives the direction, yes, exactly! The induced emf opposes the change in flux that produces it, so the induced current flows in the direction that fights that change. This is Lenz's law, and it has its own lesson; for sizes, we just use the magnitude. Clear? And in a closed circuit of resistance R, a current flows, and I equals curly E over R, that's all. Got it?

1:47Topper's trick: three ways to change the flux

Now, a trick that toppers use. Remember, phi equals B times A times cos theta: the field, times the area, times cos theta, where theta, the Greek letter, is the angle between the field and the normal to the loop. So, in how many different ways can the flux change? There are three ways, that is correct! Way number one: the field B itself grows, or falls. Right? Way number two: the area inside the field changes, like the loop being pulled out in our puzzle. With me? And way number three: the angle theta changes, because the loop turns. Clear? So, before you differentiate anything, first spot which one is changing. This habit saves many silly mistakes. Got it?

2:34The charge depends only on the change in flux

Now, the idea that settles our puzzle. Capital delta, the small triangle, stands for the change in a quantity. The average emf, curly E with a small a v below it, is N times the modulus of delta phi, over delta t; the vertical bars mean we take the size, ignoring the sign. Okay? The average current is that emf over R, the resistance of the circuit. Right? Charge is current times time, so we multiply by delta t. What happens to delta t? It cancels out, yes, exactly! So q equals I average times delta t, which is N times the modulus of delta phi, over R, nothing more. With me? Look, there is no delta t in q at all. The charge depends only on how much the flux changes, not on how fast. So, back to our puzzle: slowly or twice as fast, the same charge flows. Clear? So then, pulling faster changes nothing at all, does it? Oh no, careful, that is the trap here! Twice as fast means the flux changes twice as quickly, so the emf doubles, and the current doubles too. Only the charge stays the same. Got it? And the heat doubles as well. Double the current makes the power, I squared times R, four times bigger, but only for half the time. Four times a half is two. Okay?

4:04Example 1: average emf as B rises from 0.1 T to 0.5 T

Now our first example. We have a coil of one hundred turns, and area zero point zero one square metres; the small two on the m means squared. It is perpendicular to a field that rises from zero point one tesla, the capital T, to zero point five tesla, in zero point zero two seconds. Find the average emf. Pause the video and try it first. Done? Okay, let's check. Which of our three ways is the flux changing here? Only the field B changes, that is right! So delta B equals zero point five minus zero point one, which gives zero point four tesla. Okay? Now, many students put in zero point five tesla, the field at the end. Is that right? That is wrong, and it is a common trap! The emf comes from the change in the field, delta B, not from the field itself. So the average emf is N times A times delta B over delta t, which is one hundred times zero point zero one, times zero point four over zero point zero two. Right? One hundred times zero point zero one is just one, and zero point four over zero point zero two gives twenty. So the average emf is twenty volts, capital V for volt. Clear?

5:18Example 2: coil flipped 180 degrees, emf and charge

Now a harder one, with a trap inside it. We have a coil of fifty turns, area four square centimetres, and resistance ten ohms; the Greek capital omega is the ohm. It is perpendicular to a field of zero point five tesla, and it is flipped through one hundred and eighty degrees, the small circle, in zero point one seconds. Find the average emf, and the charge that flows. Pause the video and try it first. Done? Okay, let's check. First, four square centimetres is four times ten to the power minus four square metres. With me? Now, here comes the trap. After the flip, the field and the area are the same. So the flux has not changed, and the emf is zero, right? Oh no, careful, the flux has changed its sign! The flip turns the normal the other way, so theta goes from zero to one hundred and eighty degrees; the arrow means goes to. Cos theta goes from plus one to minus one, so the flux linkage goes from plus N times B times A, to minus N times B times A, okay? So the modulus of the change in flux linkage is two times N times B times A, not zero, and not just N times B times A, as many students write. Two times fifty times zero point five, times four times ten to the power minus four, gives zero point zero two weber, the W b on screen. Right? The average emf is zero point zero two over zero point one, which gives zero point two volts. Clear? Now, the charge: q equals the change in flux linkage over R, so zero point zero two over ten. That gives two times ten to the power minus three coulombs, capital C, which is two millicoulombs, written m C on screen. And look, the time, zero point one seconds, never entered the charge. Flip it slowly or quickly, and the same two millicoulombs flow. That solves our opening puzzle completely. Got it?

7:16Quick revision: emf, three ways flux changes, charge flow

So, quick revision. First, curly E equals minus N times d phi over d t, and only a changing flux induces an emf. Second, phi equals B times A times cos theta, so the field, the area or the angle can change. Third, q equals N times the modulus of delta phi, over R, independent of the time taken, while the emf and the heat are not. Now go and try the practice problems on this page. All the best!