Set 2

Transcript

0:00Introduction

Hello everyone! Here is a small puzzle. The magnetic field of a wave points along z, and inside its sine bracket there is a plus sign, plus x over fifteen. So the wave travels along plus x, right? Hold that thought. And which way does its electric field point? This comes up often in JEE Main. So, here is the plan for Set two. Three quick tools first, then two full problems. Let's go.

0:29Tool 1: the speed and the two amplitudes

So, tool one. In the figure, the wave travels along x. The orange E, with an arrow on top for a vector, is the electric field, along y; the dark blue B, the magnetic field, is along z; and lambda is the wavelength. Okay? Now the speed: c equals omega over k, equals nu times lambda. Omega, the curly w, is the angular frequency, k is the wave number, and nu, like a v, is the frequency. In vacuum, c is three times ten to the power eight metres per second, m s to the power minus one. Right? Now, E naught and B naught, with a small zero below, are the amplitudes of the two fields. Is E naught equal to c times B naught, or B naught equal to c times E naught? E naught equals c times B naught, yes, exactly! That is in vacuum, as the first line says. Writing it the other way round is a classic trap. Clear? Now one from this set. In vacuum, E naught is twenty seven volt per metre, written V m to the power minus one. So B naught is twenty seven, over three times ten to the power eight, which gives nine times ten to the power minus eight tesla; capital T means tesla. Got it?

1:50Tool 2: the directions of vec E, vec B and travel

Now tool two: directions. The figure says E is perpendicular to B, and both to the travel; the upside down T means perpendicular. And E cross B, the vector product, points along the travel, n hat, the unit vector with a little cap. In the figure, y cross z gives x. Okay? Turn this around: E is parallel to B cross n hat; the two upright lines mean parallel. Keep the order, B first, then n hat. Right? Now one from the set: the wave travels along plus z, with B along minus x. Which way is E? With i hat, j hat and k hat the unit vectors along x, y and z, what is minus i hat, in brackets, cross k hat? It is plus j hat, very good! Since i hat cross k hat is minus j hat, the minus sign makes it plus j hat. So E points along plus y. This sign is easy to lose. Clear?

2:55Tool 3: energy, intensity and the push of light

Now tool three: energy. Small u is the energy density, in joule per cubic metre, and the angle brackets mean its average over a cycle. With epsilon naught, the Greek letter with a small zero, the average u is one half epsilon naught E naught squared. It is shared equally by the two fields, one quarter epsilon naught E naught squared each, on average. Okay? The intensity, capital I, is the average energy per second crossing each square metre held face-on to the wave, in watt per square metre. So I equals the average u times c, which is one half epsilon naught E naught squared, times c. Got it? Light also pushes on what it hits. In the figure, the left surface absorbs the light, and the middle one, a perfect mirror, sends it straight back. Which is pushed harder? The mirror is pushed twice as hard, yes, exactly! The light's momentum is reversed. So, straight on, the pressure, small p in the figure, is I over c for a perfect absorber, and two I over c for a perfect reflector. The angled mirror on the right, at angle theta, is for another day. With me? Now one from the set: nine hundred watt per square metre falls normally on a perfect absorber. Nine hundred, over three times ten to the power eight, gives three times ten to the power minus six pascal, capital P, small a. That is tiny, isn't it?

4:28Example 1: average EM energy in a 2 litre vessel, E0 60 V/m

Now, Example one. Here, a wave with E naught equal to sixty volt per metre fills a vessel of volume two litres. Find the average electromagnetic energy in it. Pause the video and try it first. Done? Okay, let's check. Some students take the quarter from tool three, and write one quarter epsilon naught E naught squared, right? No, no, careful, that quarter is for one field alone! We want both fields, so one half. With epsilon naught as eight point eight five times ten to the power minus twelve, it is one half, times that, times sixty squared. Okay? Sixty squared is three thousand six hundred, so the average u is one point five nine times ten to the power minus eight joule per cubic metre, J m to the power minus three. Clear? Now capital V, the volume. Oh no, careful, do not just multiply by two! One litre is ten to the power minus three cubic metre, so two litres is two times ten to the power minus three, m cubed. With me? So capital U, the energy, is the average u times the volume. One point five nine times two is about three point two, so U is about three point two times ten to the power minus eleven joule; the wavy equals sign means approximately. Got it?

5:56Example 2: travel direction and E field from a given Bz

Now the harder one, our puzzle. In vacuum, B z, the B with a small z below, equals five times ten to the power minus eight, times sine of, two times ten to the power seven, times t, plus x over fifteen, in tesla. Find the direction of travel, and write the electric field. Pause the video and try it first. Done? Okay, let's check. Omega, with t, is two times ten to the power seven, and k, with x, is one over fifteen. So omega over k is two times ten to the power seven, times fifteen, which gives three times ten to the power eight metres per second, that is c. With me? Now the direction. There is a plus sign inside the bracket, so it travels along plus x, right? No, no, wait, that is the trap from our puzzle! Look at the sign between omega t and k x: minus means plus x, and plus means minus x. Here it is plus, so the wave travels along minus x, and n hat is minus i hat. Clear? Now, E naught equals c times B naught. Three times ten to the power eight, times five times ten to the power minus eight, gives fifteen volt per metre. Okay? Now the second half of our puzzle. B is along plus z, and the wave goes along minus x. Which way does E point? Use tool two: E is parallel to B cross n hat, that is, k hat cross minus i hat. Since k hat cross i hat is plus j hat, this gives minus j hat. So E points along minus y. Got it? So E y, with a small y below, equals minus fifteen, times sine of the same bracket, in volt per metre; same bracket, because E and B are in phase. Puzzle solved: the plus sign sends the wave along minus x, and E along minus y. Well done if you got both!

8:05Quick revision: E0 = cB0, E along B x n, sign rule, I/c

So, quick revision. E naught equals c times B naught, and E is parallel to B cross n hat. For the sign between omega t and k x, minus means plus x, and plus means minus x. The average u is half epsilon naught E naught squared, I is that times c, and the pressure is I over c for an absorber, two I over c for a reflector. Now try all twelve on this page, in twenty minutes. All the best!