The Wave Equations

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Here is the electric field of a wave: E equals E naught, times sine of, k x plus omega t. There is a plus sign in the middle. So this wave travels in the plus x direction, right? Hold that thought for a minute. Today we will read everything from one such line: the wavelength, the frequency, the speed, the direction, and even whether the wave is in vacuum. This comes up often in JEE, so let's begin.

0:33A plane wave travelling along +x

So, here is a plane electromagnetic wave travelling along plus x. E with a small y below it is the electric field along the y axis. E y equals E naught, times sine of the bracket, k x minus omega t. E naught, the E with a small zero, is the amplitude. Here x is the position along the axis, t is time, omega is the Greek letter like a curly w, and k we will meet in a moment. Okay? Now the magnetic field: B z equals B naught, times sine of, k x minus omega t. B with a small z below it points along the z axis, and B naught is its amplitude. See, the bracket is exactly the same, so the two fields rise and fall together. Right? Now, k is the wave number: k equals two pi over lambda. Pi is the familiar Greek letter, and lambda, the Greek letter like an upside down y, is the wavelength. Clear? And omega is the angular frequency: omega equals two pi nu, where nu, the Greek letter like a v, is the ordinary frequency. Got it? Now you know omega, and you know k. How do you get the speed of the wave from them? Just divide, yes, exactly! The speed c equals omega over k. Put in the two formulas, and the two pi cancels, leaving nu times lambda. Remember omega over k; it becomes a neat trick soon. With me?

2:02Which way does the wave travel?

Now, the direction. With k x minus omega t, or omega t minus k x, the wave travels along plus x, okay? And why is that? Follow one crest: as time t grows, x must also grow to keep the bracket the same. So the crest moves forward. Now back to our puzzle. The bracket k x plus omega t has a plus sign, so the wave goes along plus x, right? No, no, careful, that is the trap! With k x plus omega t, the wave travels along minus x. Follow the crest again: as t grows, x must now decrease to keep the bracket the same. So our puzzle is solved: the plus sign sends the wave the other way. Got it? So, to remember it, look only at the sign between k x and omega t. Minus means plus x, and plus means minus x.

2:57How E and B are related

Okay? Now, how are the two fields related? In vacuum, E naught over B naught equals c, the speed of light. With me? Now look at the middle: E over B, the fields at some moment. Is this ratio c only at the peaks, or at every moment? At every instant, yes, exactly! The fields are in phase, so E over B is c at every moment. The same holds for the rms values, the root mean square values. Right? And now a topper's trick to test for vacuum. Compute omega over k from the equation. If it is three times ten to the power eight metres per second, written m s to the power minus one on screen, the double arrow says the wave is in vacuum. Clear? But if omega over k comes out smaller, is E naught over B naught still c? Oh no, careful, it is not! Then the wave is in a medium, and omega over k is its speed there. E naught over B naught equals that speed, not c. Got it?

4:03Example 1: vacuum test with omega/k, then the wavelength

Now our first example. The wave is described by E equals ten, times sine of, four times ten to the power six, times x, minus one point two times ten to the power fifteen, times t. The V m to the power minus one on screen is volt per metre. Is it travelling in vacuum, and what is its wavelength? Pause the video and try it first. Done? Okay, let's check. Compare it with sine of, k x minus omega t: k is four times ten to the power six, and omega is one point two times ten to the power fifteen. Okay? Now, is it in vacuum? Which quick test do we use? Yes, the trick, omega over k! One point two times ten to the power fifteen, over four times ten to the power six, gives three times ten to the power eight metres per second. That is exactly c, so yes, it is in vacuum. Right? Now the wavelength. Careful: lambda is two pi over k, not one over k. So lambda equals two pi, over four times ten to the power six. The wavy equals sign means approximately, so lambda is about one point five seven times ten to the power minus six metres. With me? And ten to the power minus six metres is a micrometre, written with the Greek letter mu and a small m. So lambda is about one point five seven micrometres, in the infrared. Easy, right?

5:27Example 2: direction, wavelength, frequency and E0 from B

Now a harder one. The magnetic field of a wave in vacuum is B y equals four times ten to the power minus seven, times sine of, one point zero times ten to the power three, times x, plus three point zero times ten to the power eleven, times t. The capital T is tesla. Find the direction of travel, the wavelength, the frequency, and E naught. Pause the video and try it first. Done? Okay, let's check. Direction first. There is a plus sign inside the bracket. Which way does this wave travel? Along minus x, very good! The sign between k x and omega t is plus, so it travels along minus x, just like our puzzle. Okay? Now the wavelength. Here k is ten to the power three, so lambda equals two pi over ten to the power three, in metres. That is about six point two eight millimetres, written m m on screen. Got it? Now the frequency. The number with t is three point zero times ten to the power eleven, so that is the frequency in hertz, right? No, no, wait, that is the trap! That number is omega, the angular frequency, not nu. Nu equals omega over two pi, which is three point zero times ten to the power eleven, over two pi. That gives about four point eight times ten to the power ten hertz, written capital H, small z, on screen. Clear? And finally E naught. The wave is in vacuum, so E naught equals c times B naught. Three times ten to the power eight, times four times ten to the power minus seven, gives one hundred and twenty volt per metre. So we got four answers from one line. Well done!

7:12Quick revision: k, omega, c = omega/k, sign rule, E0/B0 = c

So, quick revision. E y equals E naught, times sine of, k x minus omega t, with k equal to two pi over lambda, omega equal to two pi nu, and c equal to omega over k. The sign between k x and omega t gives the direction: minus means plus x, plus means minus x. And in vacuum, E naught over B naught, and E over B, equal c; check for vacuum with omega over k. Now go and try the practice problems on this page. You will find them easy. All the best!