Contents
- A Little Bit of Maxwell
- Maxwell’s Idea of the Field: Maxwell’s Equations
- Wave Goodbye
- Charged Particles in Electromagnetic Waves
- What to Remember from Lesson 13
Goals of this lesson:
- I’d like you to Understand:
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How to predict the motion of a charged particle in a uniform electric field
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How to predict the motion of a charged particle in a magnetic field
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In words, not mathematically, what electromagnetic radiation is
- I’d like you to Appreciate:
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The effort to put electricity and magnetism into a single, unifying theory
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How accelerator magnets work
- I’d like you to become Familiar With:
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The role of James Clerk Maxwell in the history of physics
A Little Bit of Maxwell
Dafty. That was his nickname as a kid at the Edinburgh Academy when he entered at the age of 10.
By the age of 14 he published his first mathematical paper on a generalization of an ellipse to include “ovals.” It was the first of three papers that he presented to the Royal Academy of Scottland…none of which were accepted for publication since the editors would not believe that someone so young could have actually done that work. He’d also published poetry before going to the university when he was 16 years old. Not ordinary.
“At school he was at first regarded as shy and rather dull. he made no friendships and spent his occasional holidays in reading old ballads, drawing curious diagrams and making rude mechanical models. This absorption in such pursuits, totally unintelligible to his schoolfellows, who were then totally ignorant of mathematics, procured him a not very complimentary nickname. About the middle of his school career however he surprised his companions by suddenly becoming one of the most brilliant among them, gaining prizes and sometimes the highest prizes for scholarship, mathematics, and English verse.” as remembered by James Tait, a classmate and friend…and a pioneer of mathematical physics.
He studied at Trinity College, Cambridge (Newton’s former home-base), graduating in 1854 at 23. By the time he was 25 years old he held the chair of Natural Philosophy (“physics” in those days) at Marischal College in Aberdeen. That’s when he fixed Saturn.
A major prize was announced to explain why Saturn’s rings were stable. If they were solid, they should have broken apart. It took him two years to show that stability could only be achieved if the rings consisted of millions of small solid particles. They would all move together and clump in the way visible from a telescope…and only truly confirmed by the Voyager spacecraft in its Saturn fly-by in 1980. It’s now believed that the debris is left over from some moon of the planet that was broken up either through collision or enormous tidal forces. Said an examiner at the time of Maxwell’s submission, “It is one of the most remarkable applications of mathematics to physics that I have ever seen.”
A Voyager 2 spacecraft saturn fly-by image of some of the rings. Voyager 1 is now leaving the solar system, traveling at a speed of $3\times$ the sun-earth distance per year.
He married the daughter of the Principal of Marischal College, Katherine Dewar in 1859 and in 1860 moved to the chair of Natural Philosophy at King’s College in London and during the six years he was there he did his most important work on electromagnetic theory, the subject of this lesson. He had read Faraday’s notebooks around 1855 and corresponded with the grand old man. But nobody anticipated what he’d do with Faraday’s ideas.
In 1862 he wrote, “We can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.” And in 1873 he wrote up his research in this area which was the enunciation of the four, fundamental Maxwell’s Equations that put everything together and form the basis of modern physics and relativity theory. It is a high-altitude summary of this work that interests us in this lesson.
Maxwell also solved the problem of color vision and laid the foundations for the statistical basis of modern thermodynamics. He reluctantly found himself in the Chair at Cambridge University where he was the first Cavendish Professor of Physics in 1871. He designed the Cavendish Laboratory – in which we’ll see many 20th century milestones reached.
Unconscionably he became deadly ill with an abdominal cancer and died at the age of 48 in 1879. James Clerk Maxwell is always regarded as among the class of Newton and Einstein. Einstein wrote of Maxwell:
“[his work was the] most profound and the most fruitful that physics has experienced since the time of Newton.”
That he was as funny and companionable, as well as considerate as a supervisor rounds out a picture of one of the Big Three as being the most normal and highest quality individual among them. In these ways, he was perfectly matched for collaboration with Faraday, about whom nobody would say unpleasant things either – except about his presumed atrocious ideas. James Maxwell was indeed, Michael Faraday’s hero.
Maxwell’s Idea of the Field: Maxwell’s Equations
There are two videos, which are somewhat legacy. Section numbers are not up to date, but the physics hasn’t changed.
Think of them as lectures and start and stop them in order to absorb the information as it’s important for what follows. You’ll want to increase the size!
From this first one
VIDEO CONTENT (28 minutes), part 1:
Maxwell_fields_1
VIDEO CONTENT (28 minutes) part 1: Maxwell_fields_1
you will appreciate the origin of electromagnetic (E&M) waves – all around you, all the time – as self-generating changing $E$ fields creating changing $B$ fields and changing $B$ fields creating changing $E$ fields.
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Wave Goodbye
While the idea of an electromagnetic wave will change as we move into the late 20th century, there are some basic notions of waves that we’ll need. First doesn’t it make sense that we can think that all substances in the macroscopic world are either particles – or collections of particles – or waves.
Let’s think in terms of basics: the obvious features that distinguish particles and waves. It’s useful to think of idealized examples: an ideal particle will be one that’s infinitesimally small while an idealized wave extends in infinite directions and is perfectly repeating.
So, what is the most characteristic feature of a particle? That’s easy: Particles have a place, a definite location in space – it’s here. And furthermore, when it’s here, it’s also not simultaneously there. But a wave is everywhere, all the time. You can’t get much more different than that. It sounds almost like a distinction that only Aristotle could make, even if true.
But there are also tenuous similarities. Particles carry kinetic energy and transmit it through collisions with other particles. So too a wave is a disturbance that transmits energy. The tremendous destruction of a tsunami is terrible evidence of how energy can be imparted by waves. But the difference here is that the actual constituents of the wave don’t themselves translate but they stay put. This figure shows a finite disturbance like in a guitar string.
If you pluck a stretched string, you’ll create a disturbance that will actually move down the length of the string at a predictable speed where it would stop, or reflect and come back. If the string is attached to a bell clapper, it would ring. Obviously, energy and momentum have been transferred from your fingers through the string, sufficient to ring the bell.
Waves transmit energy.
It will be useful to think of waves as simple “sine waves.” That’s a simplification.(We will see later that any such non-repeating shape can actually be built up by a set of infinite sine waves of different wavelengths.) It’s also useful to contrast two kinds of waves: Longitudinal and Transverse.
- Longitudinal Wave: A wave in which the disturbance is along the direction of motion.
- Transverse Wave: A wave in which the disturbance is perpendicular to the direction of motion.
Longitudinal waves are compressional, like a slinky toy and appear in nature most readily in the propagation of sound. When a noise is made, the noisemaker vibrates and leaves a momentary compression or rarefaction in the surrounding air – a local high or low density region. This back and forth of high and low densities moves outward as the propagating sound wavefront. Actual molecules of the air don’t follow the wave all the way to your ear – local air molecules move and affect adjacent air molecules and that hand-off-disturbance is what propagates.
You eventually hear the sound, because that disturbance eventually reaches your ears and bangs on your eardrum like…a drum. Here is a cartoon showing two positions of a compressional disturbance along a spring-like substance.
Transverse waves are different in that the disturbance is not in the direction of the propagation, but perpendicular to it (that’s the definition of “transverse”). Water waves are the simplest example. If you toss a stone into a lake, the disturbance is the water going up and down but the wave propagates “outward” from the source in concentric circles. Again, the actual molecules of the water don’t move outward with the wave, the water molecules move up and down and affect adjacent water molecules through the tension in the water’s surface – rubber ducky just bobs up and down, he doesn’t follow the wave. This is a typical, infinite transverse wave. This figure shows a typical (infinite) transverse wave.
Wave Parameters
We can characterize a moving sine wave with only a few parameters, which are are familiar from everyday life: frequency, wavelength, and amplitude. Notice that a wave is oscillating in space – you see the water wave undulate where the peaks are all in a pattern outward from the disturbance. But also a water wave oscillates in time where each point on the surface of the water is rising or falling in rhythm with all of the other points on the surface. That means we could plot the wave as either a pattern in space or time and the functional description of a wave would contain $x$ and $t$ variables.The next two figures show these two circumstances with three important features of waves indicated on each.
The two most obvious parameters in the space picture are:
- wavelength, which is the distance in space units between any two equivalent values of the disturbance along the length of the wave (using the Greek letter, lambda $\lambda$) and
- amplitude, $A$, the maximum disturbance which can be a length (like the height of water waves) or related to other measurable parameters like the value of pressure or light intensity or other not-so-obvious quantities like for electric and magnetic fields.
In the time picture there are also two obvious parameters:
- period, usually represented as $T$—which is the time that it takes for any same value of the amplitude to repeat (which is measured in seconds) and
- amplitude, the same amplitude as in the space picture!
The most common wave parameter – it’s on every radio dial – is the frequency which is the rate at which the wave repeats: the number of repetitions per unit time which has the units of $s^{-1}$ or “per second.” This is given the name Hertz (Hz), after the great German physicist who first detected electromagnetic waves, Heinrich Hertz (1857-1894). House electrical current in the United States is a sinusoidal shape (alternating current, or “AC”) with a frequency of 60 Hz, or 60 cycles per second.
I’ll use the symbol $f$ for frequency, although it’s also commonly represented as the Greek letter $\nu$. (I don’t want to create any confusion with $\nu$for frequency and $v$ for velocity.) Obviously, if the rate that the wave changes is $f$ (“cycles per second”)and the time interval between the changes is $T$ (“seconds per cycle”), then:
The space and time representations of the pictures of waves are connected by the actual speed of the wave (which makes sense since speed connects space and time!). This connection is the important relation ($v$ is velocity!),
Here are all of the parameters and their relations:
The Speed of a Wave
The speed of a wave depends on the medium, its density, temperature, structure, and so on. But, to first approximation, the wave speed doesn’t depend on the wavelength or the frequency (or the amplitude), so if the frequency goes up for some wave (like sound) because the speed stays the same, the wavelength goes down. High frequencies mean smaller wavelengths, and visa versa.
Electromagnetism is special. And this will be the rub for a really young Albert Einstein. The speed of electromagnetic waves in a vacuum is always (Here “always” will become a surprise.):
The symbol $c$ is reserved for that very special speed of light.
Wait. You’re always doing that: hinting at something to come.
Glad you asked. Yup. It’s a narrative trick, I guess. So many properties and features of nature were completely upended, that it’s fun to get you settled into something that’s comfortable and sensible and then rhetorically pull the rug out from under you later.
An Example: Wave Properties in Everyday Life
The Question: What are the wavelengths of: the lowest C on a piano (32.7 Hz); WKAR AM radio radiation (871 kHz on your radio dial)? Assume that the speed of sound in air is 341 m/s and that the speed of light in a vacuum is $3\times 10^{8}$ m/s.
We can easily use Eq. \eqref{wave_properties} twice to find these.;
Our concern will be with the “electromagnetic spectrum” and here it is in its full glory:
Electromagnetic waves all move at $c$ so they can be characterized by either their frequency or their wavelengths. Notice that I wrote “rays” and “waves” which correspond to their everyday usage. This respects the history of their discovery as well see.
Wait. There you go again.
Tired of you asking. $\star$ crickets $\star$
Charged Particles in Electromagnetic Waves
An electromagnetic wave (E&M wave) is a transverse wave with two coupled amplitudes of an electric and magnetic field vector traveling at a single speed, that of light: $c=3 \times 10^8$m/s. Your cell phone signal? E&M waves. The light through your window? E&M waves. The warmth you feel on (and radiate from) your skin. Yup, E&M waves. Back and forth. We’ll see that even the teenage Einstein found a contradiction here that in part stimulated his venture into Special Relativity.
The second piece of the story is next. Here we’ll put together the manner in which charged particles are stimulated by passage through electric and magnetic fields. With this, we’re ready to build a particle accelerator.
VIDEO CONTENT (21 minutes), part 2:
Maxwell_fields_2
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What to Remember from Lesson 13
Waves
The wavelength ($\lambda$), frequency ($f$), and speed ($v$) of a wave are all related through a simple model:
Electromagnetism
James Clerk Maxwell found mathematically that
- a changing magnetic field would induce changing electric field and that
- changing electric field would induce changing magnetic field.
And furthermore, they are coupled together into a single entity that moves in time – the electromagnetic wave. He explained all of electricity and magnetism in this way and further could account for all of Faraday’s experiments.
The speed of electromagnetic waves is always
- $c= 3 \times 10^{8}$ m/s
The Lorentz Force describes how charged particles react to electric and magnetic fields. It has two pieces:
- A charged particle in an electric field is accelerated along the field lines (if a positive particle) or away from the field lines (if a negative particle).
- A charged particle will bend in the presence of a magnetic field if it is moving. The direction of the force on a charged particle is perpendicular to the field and is found with your right hand curing your fingers through the velocity vector and then into the direction of the magnetic field. Your thumb then will point in the direction of the force that a positively charged particle will experience or the opposite direction of the force that a negatively charged particle will experience.