Note to the teacher: The formula E = mc 2 is discussed in detail in the Background Material “Conservation of Energy—Revisited”. Here, we discuss three other relevant formulae, one that the students may have some knowledge of, and two they have probably never seen before: #1E = hf #2E =  mc 2 #3

#1 E = hf

Hello.

As the CEO of a large corporation, this formula seems a little difficult to me, and I am feeling a slight pain in my head from it.

λ

λ λ

0.00 seconds

0.25 seconds

0.50 seconds

f = 1 cycle per 0.50 seconds = 2 cycles per second = 2 Hz

E = hf THE CONVERSION FACTOR h IS CALLED PLANCK’S CONSTANT, AND IT HAS AN EXTREMELY SMALL VALUE: h = 6.63 × 10  34 joule seconds (Js) WE CAN ALSO EXPRESS IT IN DIFFERENT UNITS: h = 4.14 × 10  15 electron volt seconds (eVs)

IF WE WANT TO KNOW THE WAVELENGTH OF THE PHOTON, WE USE A SLIGHTLY DIFFERENT VERSION OF THE FORMULA: WHERE c IS THE SPEED OF LIGHT: 3.00 × 10 8 meters per second (m/s) THIS VERSION OF THE FORMULA TELLS US THAT THE GREATER THE WAVELENGTH OF THE PHOTON, THE LESSER ITS ENERGY.

f INCREASING ENERGY THESE FORMULAS ALSO ALSO REVEAL THAT THE ENERGIES IN THE ELECTROMAGNETIC SPECTRUM INCREASE AS WE GO FROM SMALLER TO LARGER FREQUENCIES, AND DECREASE AS WE GO FROM SHORTER TO LONGER WAVELENGTHS.

#2 E =  mc 2

As the CEO of a large corporation, this formula seems quite difficult to me, and has begun to make my head hurt a lot.

YOU HAVE ALREADY SEEN THIS FORMULA: E = mc 2 THIS ALLOWS US TO CONVERT MASS (m) TO ENERGY (E). ITS SLIGHTLY REARRANGED FORM: CAN BE USED TO CONVERT ENERGY TO MASS.

E = mc 2 GIVES WHAT IS CALLED THE “REST ENERGY”. THIS MEANS EXACTLY WHAT IT SAYS: IT IS THE TOTAL ENERGY OF A MASS THAT ISN’T MOVING. IF WE HAVE A MASS THAT IS MOVING WE NEED TO CHANGE THIS FORMULA A LITTLE…

MEET E =  mc 2 THIS FORMULA GIVES THE TOTAL ENERGY OF A MASS THAT IS IN MOTION. IT APPEARS TO BE IDENTICAL TO E = mc 2 BUT WITH SOMETHING EXTRA WHAT IS  ? ( BESIDES BEING A GREEK LETTER )

AAAAHHHHHH!

THIS EXPRESSION IS KNOWN AS THE GAMMA FACTOR.

IT’S JUST A FRACTION —

GONE MAD!

IT’S JUST A FRACTION — GONE MAD! Numerator

Denominator IT’S JUST A FRACTION — GONE MAD!

Another fraction, with the numerator and denominator squared IT’S JUST A FRACTION — GONE MAD! Numerator Denominator

Square root of the denominator IT’S JUST A FRACTION — GONE MAD! Another fraction, with the numerator and denominator squared Numerator Denominator

v is the speed at which the mass is moving c (as you already know) is the speed of light Also, notice that we can write as

When we divide a very large number into a much smaller one, the result itself is very small. Since the speed of light is huge, if we were to take an everyday speed and divide by c, we would get a very small value. If we were to then square this, the result is even smaller. Take that and subtract from one? Well that’s going to be extremely close to ONE. Under the square root (in the denominator), we have

To prove to yourself that this is true, take a speed v = 80 km/h (≈ 22 m/s) and divide by c (= 3.0 ×10 8 m/s) [Answer: ] If we now square that answer, we get: Finally, = which, for most purposes, can be written simply as: 1

So, for “everyday speeds”, ≈ 1 AND ≈ = 1

NOW, as v gets larger, so does Thus, will become smaller, and will become larger.

To prove to yourself that this is true, take a speed v of 99.5% the speed of light (0.995c): Then,= (0.995) 2 ≈ 0.99 and ≈ 0.01 So, ≈ = 10

AND WHAT HAPPENS WHEN V = C ? Then = 0 AND = WHICH IS NOT ALLOWED IN MATHEMATICS.

AND IT IS NOT ALLOWED IN NATURE, EITHER. NOTHING THAT HAS MASS CAN MOVE AT THE SPEED OF LIGHT! As an exercise, examine what happens to the the gamma factor if we use a speed which is GREATER than the speed of light.

 0.1c0.2c0.5c0.3c0.6c0.4c1.0c0.7c0.9c0.8c v THIS GRAPH SHOWS HOW THE GAMMA FACTOR VARIES WITH SPEED: The gamma factor remains very close to 1, even for speeds (v) of one-half the speed of light. At speeds greater than 0.95c, the gamma factor starts to become very large, very quickly.  -factor vs. speed

IN ADDITION TO THE GRAPH, THIS TABLE CAN BE USED TO FIND THE GAMMA FACTOR FOR VARIOUS SPEEDS

SO MUCH FOR THE GAMMA FACTOR, BUT WHAT DOES THE FORMULA E =  mc 2 ACTUALLY TELL US?

SO MUCH FOR THE GAMMA FACTOR, BUT WHAT DOES THE FORMULA E =  mc 2 ACTUALLY TELL US? When a mass is moving, it has two distinct types of energy: the energy due to its mass (the rest energy mc 2 ) AND the energy due to its motion (the kinetic energy or E k )

SO MUCH FOR THE GAMMA FACTOR, BUT WHAT DOES THE FORMULA E =  mc 2 ACTUALLY TELL US? When a mass is moving, it has two distinct types of energy: the energy due to its mass (the rest energy mc 2 ) AND the energy due to its motion (the kinetic energy or E k ) The total energy, as given by E =  mc 2, is actually the sum of these two: Total energy = rest energy + kinetic energy OR  mc 2 = mc 2 + E k

Also note that, if we are interested in finding only the kinetic energy, we can rearrange  mc 2 = mc 2 + E k and write E k =  mc 2 − mc 2 OR E k = (  − 1)mc 2 Note that for a speed of 0.995c, E k ≈ (10 − 1)mc 2 = 9mc 2. This can sometimes be useful for rough approximations.

#3

NO !!!!

Don’t worry. This will in no way prevent me from doing my job. Don’t worry. This will in no way prevent me from doing my job.

We do not expect you to be able to understand this formula at this stage! For that, you will have to wait until you have a great deal more knowledge of mathematics. * We have shown it to you here because of its enormous significance in the history of human understanding about the existence of antimatter. * If students are interested, there is a simplified explanation of this formula in the book Antimatter by Frank Close (2009, Oxford University Press) that may be suitable for higher level students who have some knowledge of matrices.

It is called the Dirac Equation, named for the physicist who formulated it, Paul A. M. Dirac.

This he did in 1928, at the age of 26.

You are familiar with the idea that when you solve an equation, there are sometimes more than one solution. For instance, when we take the square root of a number, you know that there are exactly two solutions. For example, = +4 or −4 Without going into detail (which would be far beyond the scope of this module), it turns out that when one solves this equation, there are two possible solutions… This is a mathematical reality, and we can’t escape it. When we apply mathematics to the study of nature, we also sometimes get two (or more) possible solutions to our formulas. Very often, only one of these solutions will make sense in nature, and we may choose to disregard the other.

The brilliance of Paul Dirac was that he insisted that both of the solutions of his equation had a basis in reality: one which made sense for matter, and another that made sense only if there was another type of material—antimatter. So, the existence of antimatter was predicted (using mathematics) before anyone had even dreamt of such a thing. This plaque was unveiled in Westminster Abbey in 1995 to commemorate Paul Dirac. This is the first equation to appear in the Abbey and celebrates Dirac's achievements as one of the founding fathers of quantum physics. If you would like to learn more about Paul Dirac, click on the link below: