Energy Density

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. In a light wave in vacuum, the magnetic field B is the electric field E divided by c, and c is three times ten to the power eight metres per second. So in tesla, B is a really tiny number. Does the magnetic field then carry almost no energy? Keep your answer in mind. Today we will see how a wave shares its energy between its two fields. This comes up often in JEE, so let's begin.

0:31Energy stored in each field

So, a wave stores energy in its fields. The letter u is the energy density, the energy stored per unit volume. Okay? The u with a small E below it is the electric energy density, and u E equals one half, times epsilon naught, times E squared. Epsilon naught, the Greek letter epsilon with a small zero, is the permittivity of free space. Right? And the u with a small B below it is the magnetic energy density, and u B equals B squared, over two mu naught. Mu naught, the Greek letter mu with a small zero, is the permeability of free space. Clear? And the unit on screen, J m to the power minus three, is joule per cubic metre. Got it?

1:17The energy is shared equally

Now, back to our puzzle. In vacuum, E equals c times B, where c is the speed of light. And c squared equals one over, mu naught times epsilon naught. With me? So let me ask you this. Put B equals E over c into u B, the magnetic energy density. Is it really smaller than the electric one? Put B equals E over c into B squared over two mu naught, and we get E squared, over two mu naught c squared. But mu naught times c squared is one over epsilon naught, so this is epsilon naught E squared, over two. That is exactly u E, the electric energy density! So no, it is not smaller at all. Okay? So u E equals u B, at every point and every instant, and the energy is shared equally. See, B is tiny, but B squared is divided by mu naught, which is also very small, so they balance out. That answers our puzzle, right? Now, if each field holds an equal share, what is the total energy density? Yes, just add them, so u equals u E plus u B, which is twice one half epsilon naught E squared. So u equals epsilon naught E squared, or B squared over mu naught. Clear?

2:40Averaging over one cycle

Now, in a real wave, E equals E naught, times sine of, k x minus omega t. E naught, the E with a small zero, is the amplitude, k is the wave number, and omega, the Greek letter like a curly w, is the angular frequency. The angle brackets mean the average over one cycle, and sine squared averages to one half. Okay? So, tell me this. u E is one half epsilon naught E squared, so what is its average over one cycle? Yes, exactly, one quarter epsilon naught E naught squared! E squared is E naught squared times sine squared, and that sine squared averages to one half. The magnetic part has the same average, since the two are equal at every instant. Got it? Now add the two averages. The average total energy density is one half epsilon naught E naught squared, which also equals B naught squared over two mu naught. B naught, the B with a small zero, is the magnetic amplitude. Right? At the instant when E equals E naught, its biggest value, what is the energy density? It is epsilon naught E naught squared, the peak, which is twice the average. This is a common trap: students use this peak value as the average, and get double the answer. For the average, keep the half. Clear? Now, sometimes the question gives E rms, the root mean square value, written with the small letters r m s. Is the average then one half epsilon naught E rms squared? No, no, careful, there is no half here! E rms squared is already E naught squared over two. So the average is simply epsilon naught E rms squared. With me?

4:32How fast does u change at a point?

Here is one more lovely idea. Since u equals epsilon naught E squared, u is proportional to sine squared of, k x minus omega t. This sign, like an open sideways eight, means proportional to. Okay? Now write sine squared as one half, times the square bracket, one minus cosine of twice the bracket, k x minus omega t. Look at that two inside the cosine. What does it tell you? Yes, exactly, at a fixed point the energy density oscillates at twice the frequency of the wave. The field rises and falls once per cycle, but its square peaks twice, at the crest and at the trough. Right?

5:14Example 1: average energy density for E0 = 100 V/m

Now, our first example. Here, a wave has E naught equal to one hundred volt per metre, written V m to the power minus one. Find its average energy density. Pause the video and try it first. Done? Okay, let's check. The question says average, so we use one half epsilon naught E naught squared, and not epsilon naught E naught squared, which is the peak. Clear? Put in the numbers, and we get one half, times eight point eight five times ten to the power minus twelve for epsilon naught, times one hundred squared. Okay? One hundred squared is ten to the power four, and half of eight point eight five is four point four two five. So the answer, with the wavy equals sign for approximately, is about four point four times ten to the power minus eight joule per cubic metre. Forget the half, and you get double, which is the trap answer.

6:15Example 2: E0 = 30 V/m, average energy in a 2 cm2 tube

Got it? Now a harder one. Here, a wave with E naught equal to thirty volt per metre fills a tube of cross-section two square centimetres, written c m squared, and length fifty centimetres. Find the average energy in the tube. Pause the video and try it first. Done? Okay, let's check. The average energy density is one half epsilon naught E naught squared. Thirty squared is nine hundred, so it is one half, times eight point eight five times ten to the power minus twelve, times nine hundred. Okay? That works out to about three point nine eight times ten to the power minus nine joule per cubic metre. Right? Now the volume, and here is the trap. What is two square centimetres in square metres? It is two times ten to the power minus four square metres, not ten to the power minus two, because one centimetre is ten to the power minus two metre, and we square it. Fifty centimetres is zero point five metre. So the volume is ten to the power minus four cubic metres, written m cubed. Clear? Finally, energy equals average energy density times volume. Three point nine eight times ten to the power minus nine, times ten to the power minus four, gives about four point zero times ten to the power minus thirteen joule, the capital J on screen. Remember our puzzle: half of this energy is electric, and half is magnetic. Well done!

7:52Quick revision: uE = uB, average, peak, twice the frequency

So, quick revision. u E, one half epsilon naught E squared, equals u B, B squared over two mu naught, so the energy is shared equally, and the total is epsilon naught E squared. The average is one half epsilon naught E naught squared, or epsilon naught E rms squared, and the peak is twice the average. And at a fixed point, u oscillates at twice the frequency of the wave. Now go and try the practice problems on this page. All the best!