1. OH NO!!
FORMULAS!

2. The formula E = mc2 is discussed in detail in the Background Material
Note to the teacher:
The formula E = mc2 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:
#1 E = hf
#2 E = gmc2 #3

3. #1
E = hf

5. Hello.

6. 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.

7. The photon can be considered as a tiny packet of energy.
For example, Light is a stream of photons.

8. And, although it defies common sense (and isn’t easy to imagine),
the photon has the characteristics of both…

9. And, although it defies common sense (and isn’t easy to imagine),
the photon has the characteristics of both…
A particle

10. And, although it defies common sense (and isn’t easy to imagine),
the photon has the characteristics of both…
A particle

11. And, although it defies common sense (and isn’t easy to imagine),
the photon has the characteristics of both…
A particle
&
A WAVE

12. In fact, we believe that everything exhibits this double nature BUT
that it only really reveals itself in the realm of the very small. This is known as “wave-particle duality”.
every particle has both a wave and a particle nature.

13. WE BRING up this concept AT THIS TIME SO THAT WE CAN DESCRIBE SOME
properties OF the PHOTON
USING TERMINOLOGY USUALLY
reserved for WAVES,
SUCH AS…

14. (the distance measured in meters between successive wave-tops)
WAVELENGTH (λ)
(the distance measured in meters between successive wave-tops)

15. (the distance measured in meters between successive wave-tops)
WAVELENGTH (λ)
(the distance measured in meters between successive wave-tops) λ

16. (the distance measured in meters between successive wave-tops)
WAVELENGTH (λ)
(the distance measured in meters between successive wave-tops) λ λ

17. (how many times per second a cork, for example,
FREQUENCY (f)
(how many times per second a cork, for example,

18. (how many times per second a cork, for example,
FREQUENCY (f)
(how many times per second a cork, for example,

19. (how many times per second a cork, for example,
FREQUENCY (f)
(how many times per second a cork, for example,
goes through a cycle of moving up, down, and back up again as waves pass by, measured in number of cycles per second OR Hertz Hz)

20. (how many times per second a cork, for example,
FREQUENCY (f)
(how many times per second a cork, for example,
goes through a cycle of moving up, down, and back up again as waves pass by, measured in number of cycles per second OR Hertz Hz)
0.00 seconds

21. (how many times per second a cork, for example,
FREQUENCY (f)
(how many times per second a cork, for example,
goes through a cycle of moving up, down, and back up again as the waves pass by, measured in number of cycles per second OR Hertz Hz)
0.25 seconds

22. (how many times per second a cork, for example,
FREQUENCY (f)
(how many times per second a cork, for example,
goes through a cycle of moving up, down, and back up again as the waves pass by, measured in number of cycles per second OR Hertz Hz)
0.50 seconds

23. (how many times per second a cork, for example,
FREQUENCY (f)
(how many times per second a cork, for example,
goes through a cycle of moving up, down, and back up again as the waves pass by, measured in number of cycles per second OR Hertz Hz)
f = 1 cycle per 0.50 seconds
= 2 cycles per second
= 2 Hz
0.50 seconds

24. E = hf THE ENERGY OF A SINGLE PHOTON IS GIVEN BY THE FOLLOWING
SIMPLE LITTLE FORMULA:
E = hf
WHERE ‘h’ is a conversion factor that allows us to convert the frequency of a photon into the photon’s energy.

25. E = hf THE ENERGY OF A SINGLE PHOTON IS GIVEN BY THE FOLLOWING
SIMPLE LITTLE FORMULA:
E = hf
WHERE ‘h’ is a conversion factor that allows us to convert the frequency of a photon into the photon’s energy.
According to this formula,
the greater the frequency (f) of a photon,
the greater its energy (E).

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

27. 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 × 108 meters per second (m/s)
THIS VERSION OF THE FORMULA TELLS US THAT
THE GREATER THE WAVELENGTH OF THE PHOTON,
THE LESSER ITS ENERGY.

28. 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.

29. #2
E = mc2

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

31. E = mc2 YOU HAVE ALREADY SEEN THIS formula:
THIS ALLOWS US TO CONVERT MASS (m) TO energy (E).
its slightly rearranged form:
Can be used to convert Energy to mass.

32. E = mc2 ISN’T MOVING. 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…

33. MEET
E = mc2
THIS FORMULA GIVES THE TOTAL ENERGY OF A MASS THAT IS IN MOTION.
IT APPEARS TO BE IDENTICAL TO E = mc2
BUT WITH SOMETHING EXTRA
WHAT IS  ?
(BESIDES BEING A GREEK LETTER)

35. AAAAHHHHHH!

36. THIS EXPRESSION IS KNOWN AS THE GAMMA FACTOR.
AAAAHHHHHH!
THIS EXPRESSION IS KNOWN AS THE GAMMA FACTOR.

37. IT’S JUST A FRACTION —

38. IT’S JUST A FRACTION —
GONE MAD!

39. IT’S JUST A FRACTION — GONE MAD!
Numerator

40. IT’S JUST A FRACTION — GONE MAD!
Numerator
Denominator

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

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

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

44. Under the square root (in the denominator), we have
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.

45. 1 To prove to yourself that this is true, take a speed
v = 80 km/h (≈ 22 m/s) and divide by c (= 3.0 ×108 m/s)
[Answer: ]
If we now square that answer, we get:
Finally, =
which, for most purposes, can be written simply as: 1

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

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

48. 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

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

50. MASS AND IT IS NOT ALLOWED IN NATURE, EITHER. NOTHING THAT HAS
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.

51. THIS GRAPH shows how the gamma factor varies with speed:
-factor vs. speed 
0.1c
0.2c
0.5c
0.3c
0.6c
0.4c
1.0c
0.7c
0.9c
0.8c v
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.

52. IN ADDITION TO THE GRAPH, THIS TABLE
Speed
(v) 
(to 3 decimal places)
c (= 100 km/h)
1.000
0.1c
1.005
0.2c
1.021
0.3c
1.048
0.4c
1.091
0.5c
1.155
0.6c
1.250
0.7c
1.400
0.8c
1.667
0.9c
2.294
0.95c
3.203
0.99c
7.089
0.995c
10.013
0.999c
22.366
0.9999c
70.712 c
IN ADDITION TO THE GRAPH, THIS TABLE
CAN BE USED TO FIND
THE GAMMA FACTOR
FOR VARIOUS SPEEDS

53. SO MUCH FOR THE GAMMA FACTOR, BUT WHAT DOES THE FORMULA
E = mc2
ACTUALLY TELL US?

54. SO MUCH FOR THE GAMMA FACTOR, BUT WHAT DOES THE FORMULA E = mc2
ACTUALLY TELL US?
When a mass is moving, it has two distinct types of energy:
the energy due to its mass (the rest energy mc2)
AND
the energy due to its motion (the kinetic energy or Ek)

55. SO MUCH FOR THE GAMMA FACTOR, BUT WHAT DOES THE FORMULA
E = mc2
ACTUALLY TELL US?
When a mass is moving, it has two distinct types of energy:
the energy due to its mass (the rest energy mc2)
AND
the energy due to its motion (the kinetic energy or Ek)
The total energy, as given by E = mc2, is actually the sum of these two:
Total energy = rest energy + kinetic energy OR
mc2 = mc2 + Ek

56. we can rearrange mc2 = mc2 + Ek and write
Also note that, if we are interested in finding only the kinetic energy,
we can rearrange mc2 = mc2 + Ek and write
Ek = mc2 − mc2 OR
Ek = ( − 1)mc2
Note that for a speed of 0.995c, Ek ≈ (10 − 1)mc2 = 9mc2. This can sometimes be useful for rough approximations.

57. #3

61. NO!!!!

62. NO!!!!

63. NO!!!!

64. NO!!!!

65. NO!!!!

66. NO!!!!

67. NO!!!!

69. 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.

70. 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.

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

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

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

74. This is a mathematical reality, and we can’t escape it.
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…
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
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.

75. So, the existence of antimatter was predicted (using mathematics)
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 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: