Production and Nature of Electromagnetic Waves

Transcript

0:00Introduction

Hello everyone! Here is a small puzzle. Picture three charged particles. The first sits still, the second moves in a straight line at a steady speed, and the third wiggles back and forth. Only some of them send out an electromagnetic wave, the same kind of wave as light. So which of them do it? Today we find out, and see what these waves look like. This topic comes up often in JEE, so let's begin.

0:27Which charges send out waves?

So, let us take the charges one by one. First, a charge at rest produces only an electric field: the E with a small arrow on top, where the arrow tells you it is a vector. Okay? Now let the charge move in a straight line with constant velocity. So tell me, does it send out a wave? It does not send out any wave. Moving with constant velocity, the charge produces steady fields: the electric field, and B with an arrow, the magnetic field. Steady means they do not change with time, so there is no wave. Right? So the answer to our puzzle is acceleration. Only an accelerated charge produces fields that change in time, and these changing fields sustain each other and travel outwards as an electromagnetic wave. The double arrow on screen means this gives a wave. Clear? Now here is a trap. Suppose a charge goes round and round a circle at constant speed. The speed never changes, so it does not radiate, right? Oh no, careful, it does radiate! In circular motion the direction of the velocity keeps changing, so the charge is accelerated. Radiation needs acceleration, and circular motion is accelerated. Got it? And what about the frequency of the wave? If a charge oscillates with frequency nu, the Greek letter that looks like a v, it radiates a wave of the same frequency nu. So the wiggling charge from our puzzle does radiate. With me? And an L C circuit radiates at a frequency nu equal to one over, two pi times the square root of L times C, where L is the inductance and C is the capacitance. The tick shaped sign on screen is the square root, okay?

2:16What the wave looks like

Now let us see what this wave looks like. In the figure, x is the direction of travel. The electric field swings along y, and the magnetic field swings along z, right? So the two fields are perpendicular to each other; the upside down T on screen is the symbol for perpendicular. And both are perpendicular to the direction of travel too. Fields that swing across the direction of travel make a transverse wave. Clear? But how do we know these waves are really transverse? Which experimental fact proves it? The answer is polarisation, yes, exactly! Only transverse waves can be polarised, and electromagnetic waves can be. So they must be transverse. Got it? Now, the two fields point at ninety degrees to each other. The small circle on screen means degrees. So the phase difference between them is also ninety degrees, right? No, no, wait, that is the trap! The angle between their directions is ninety degrees, but the phase difference is zero. The two fields oscillate in phase: they reach their peaks together, and their zeros together, as you can see in the figure. With me? Of the two fields, the electric field is responsible for the optical effects, so it is called the light vector. Okay?

3:36Speed, and what the waves carry

Now let us look at the speed. These waves need no medium at all. In vacuum, their speed c equals one over the square root of mu naught times epsilon naught. Mu naught, with a small zero below the mu, is the permeability of free space, and epsilon naught, the curly e with a small zero, is the permittivity of free space. That is three times ten to the power eight metres per second, the m s to the power minus one on screen. With me? Now tell me, red light and X-rays both travel through vacuum; which one is faster? Neither of them, they have the same speed c! In vacuum, every electromagnetic wave travels at c, whatever its frequency, so red light and X-rays are equally fast. Right? But careful, c is the speed in vacuum. From air into glass, the wave slows down. The frequency is fixed by the source, so it stays the same, and the wavelength, lambda, the Greek letter like an upside down y, gets shorter. Clear? Next, these waves carry energy and momentum, and they exert pressure too. Got it? Suppose a beam of electromagnetic waves passes between the plates of a charged capacitor. Is the beam deflected? It is not deflected, that is correct! The wave carries no charge, so electric and magnetic fields cannot deflect it.

5:00Example 1: wave frequency and wavelength from a 1 GHz charge

Okay? Now our first example. Read it with me: a charge oscillates at ten to the power nine hertz; the capital H and small z on screen stand for hertz. Find the frequency and the wavelength of the wave it radiates in vacuum. Pause the video and try it first. Done? Okay, let's check. What is the frequency of the wave? We need no formula for this! The wave has the same frequency as the oscillating charge, so it is ten to the power nine hertz. Right? Now let us find the wavelength. Lambda equals c over nu, which is three times ten to the power eight, over ten to the power nine. That gives zero point three metres; the small m on screen is metres. Easy, isn't it?

5:48Example 2: 300 m waves, frequency and LC capacitance

Now a harder one. We want to radiate waves of wavelength three hundred metres efficiently. What frequency must the oscillating charge have? And if the circuit has L equals one hundred microhenry, the mu H on screen, what capacitance is needed? Pause the video and try it first. Done? Okay, let's check. Nu equals c over lambda, which is three times ten to the power eight, over three hundred. That gives ten to the power six hertz, which is one megahertz, written M H z on screen. Okay? Now let us find the capacitance. Start from the L C formula. Square both sides and rearrange, and you get C equals one over, four pi squared times nu squared times L, clear? So now we just put in L equals one hundred, right? No, no, wait, that is the trap! Micro means ten to the power minus six. So L equals one hundred times ten to the power minus six, which is ten to the power minus four henry, the capital H on screen. Got it? Now put it all in. Nu squared is ten to the power six, squared, which is ten to the power twelve. So C equals one over, four pi squared times ten to the power twelve times ten to the power minus four, and the powers of ten combine to ten to the power eight. Right? Four pi squared is about thirty nine point five; the wavy equals sign means approximately. So C is about two point five three times ten to the power minus ten farad, the capital F on screen. Pico means ten to the power minus twelve, so that is two hundred and fifty three picofarad, the p F on screen. If you got both answers, well done!

7:31Quick revision: accelerated charges, transverse, speed c

So, quick revision. Only accelerated charges radiate, and the wave has the frequency of its source. The two fields are perpendicular to each other and to the direction of travel, and in phase, so the waves are transverse. In vacuum they all travel at c, three times ten to the power eight metres per second, and with no charge, fields do not deflect them. Now go and try the practice problems on this page. All the best!