Part C: Concept Checks

Transcript

0:00Introduction

Hello everyone! Here is a small puzzle. Light carries momentum, so sunlight falling on your hand really does push on it. So why do you never feel that push? Keep your answer in mind; Example two settles it. Today, eight quick concept checks on electromagnetic waves; such why-questions come up often in JEE, so let's begin.

0:25Check 1: why was the displacement current needed?

So, check one: why was the displacement current needed? Ampere's law links the magnetic field round a closed loop to the current, i, through any surface on that loop. Okay? In the figure, a capacitor is charging, with plus q and minus q on its plates, and i c, the i with a small c below, in the wires; we skip the graph on the right. The wire cuts a flat surface on loop one, so the current through it is i. Now bulge that surface between the plates. What current crosses it? It is zero, exactly, because no charge crosses the gap. The same loop gives two answers, so the law was incomplete; that is why the displacement current was needed. Clear? Between the plates, the electric field, the orange arrows, keeps changing. So Maxwell added i d, the label at the top of the figure, the i with a small d below, equal to epsilon naught, Greek epsilon with a small zero, times d phi E by d t; phi E, capital phi with a small E below, is the electric flux. It exists only where the electric flux changes, here between the plates while the capacitor charges. Right? And once the capacitor is fully charged, the field stops changing, so d E by d t is zero, and i d is zero too. Got it?

1:43Check 2: no medium needed, and no bending

Now check two. Sunlight crosses empty space to reach us. So why do electromagnetic waves not need a medium? The answer is in Maxwell's equations. Here a changing magnetic field B, the arrow on top meaning a vector, produces an electric field E; the double arrow means gives. And a changing E produces B, in turn. In the figure, E, in orange, points along y, and B, in dark blue, along z; they oscillate in step, at right angles to each other and to x, the direction of travel. The upside-down T signs at the top mean perpendicular, and lambda, the Greek letter, is the wavelength. Okay? So each field regenerates the other. The wave is just these changing fields, so it needs no medium, and it travels even through empty space. Clear? In vacuum, all of them travel at c, one over the square root of mu naught times epsilon naught, where mu naught is the Greek mu with a small zero, that is, three times ten to the power eight metres per second, the m s to the power minus one on screen. In a medium they are slower. With me? Now a quick one. An electromagnetic wave passes through a strong uniform magnetic field. Is its path bent? It is not bent at all: the wave has no charge, and in vacuum fields simply superpose, that is, they just add up.

3:12Check 3: what fixes the colour of light?

Right? Now check three. When light enters glass from air, its wavelength changes, but its colour does not. So which quantity fixes the colour? In the glass, the frequency, nu, the Greek letter, does not change. The speed becomes v equals c over n, with n the refractive index, and the wavelength lambda becomes lambda naught over n, where lambda naught, with the small zero, is its value in vacuum. Okay? So the colour is fixed by the frequency, the one quantity that does not change. Many students say wavelength, but that is exactly what changed! Got it?

3:49Check 4: electric force versus magnetic force

Now check four. On an electron of charge e, a light wave's electric field gives a force e times E, and its magnetic field a force e times v times B, with v the electron's speed. Yet the electric field moves the electrons of matter far more. Why? Use the wave's own relation: B equals E over c. So e v B becomes e E times v over c. Right? And electron speeds in matter are much less than c; the double less-than sign means much less than. So the magnetic force is smaller by the factor v over c. Clear?

4:24Check 5: an Earth with no atmosphere

Now check five. If the Earth had no atmosphere, would its average surface temperature be higher or lower than now? Many students say higher, since nothing would block the sunlight. But visible sunlight warms the ground, and the ground radiates infrared, which carbon dioxide, the C O two on screen with a small two below, and water vapour absorb and partly send back. This is the greenhouse effect. Okay? So with no atmosphere there is no greenhouse effect to trap that infrared, and the Earth would be colder; the answer is lower. Got it?

4:59Example 1: why a mirror feels twice the radiation pressure

Now, Example one. Why is the pressure on a perfect reflector twice that on a perfect absorber? Pause the video and try it first. Are you done? Okay, let's check. Light carrying energy, capital U, also carries momentum, small p, equal to U over c. When it is absorbed or reflected, this momentum passes to the surface, which feels a pressure. Okay? In the figure, the orange rays stop at the absorber on the left, and come straight back from the perfect reflector in the middle. In which case does the light's momentum change more? Take the absorber first. The momentum goes from p to zero, the arrow meaning goes to, so the change is just p. That gives p rad, the pressure, with rad written small below, equal to I over c, where capital I is the intensity; that is the p under the left picture. Right? Now the reflector. The momentum is reversed, from p to minus p, so the change is two p, double! So the pressure is two I over c, as under the middle picture. Clear? Both results are for normal incidence, light falling straight on; we skip the angled case on the right. So reflection reverses the momentum and doubles its change, and that is the whole answer.

6:20Example 2: why the push of sunlight is never felt

Now Example two, and it is our puzzle from the start. Why is the radiation pressure of sunlight not noticed in daily life? Pause the video and try it first. Are you done? Okay, let's check. For an absorber the pressure is I over c, and c, three times ten to the power eight, is a huge number to divide by. Okay? For sunlight, take the intensity from Chapter three, fourteen hundred watt per square metre, the W m to the power minus two on screen. Fourteen hundred over three times ten to the power eight gives about four point seven times ten to the power minus six; the wavy equals sign means about. So the pressure is roughly five times ten to the power minus six pascal, the P a on screen. Right? Now, is five millionths of a pascal small? And small compared with what? Compare it with the air around us: atmospheric pressure is ten to the power five pascal. Clear? Five times ten to the power minus six, over ten to the power five, gives five times ten to the power minus eleven. So sunlight pushes with only about five parts in a hundred billion of atmospheric pressure, far too small to feel. That settles our puzzle, very good!

7:37Quick revision: displacement current to radiation pressure

So, quick revision. Displacement current exists only where the electric flux changes; the fields regenerate each other, so no medium, and with no charge, no bending. Colour is fixed by the frequency, the magnetic force is v over c of the electric, and no atmosphere means a colder Earth. Pressure is I over c on an absorber, two I over c on a reflector, and sunlight's is only five times ten to the power minus eleven of air pressure. Now try the practice problems on this page. All the best!