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
 
ig



m   0
#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
0.50
seconds
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:
E
hc

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.
INCREASING ENERGY
f
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=g
2
mc
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 = mc2
THIS ALLOWS US TO CONVERT MASS (m) TO ENERGY (E).
ITS SLIGHTLY REARRANGED FORM:
E
m 2
c
CAN BE USED TO CONVERT ENERGY TO MASS.

E = mc2
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 = gmc2
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 g
?
(BESIDES BEING A GREEK LETTER)
g
1
1

V
2
C
2
AAAAHHHHHH!
g
1
1

V
2
C
2
AAAAHHHHHH!
g
1
1
V
2
C
2
THIS EXPRESSION IS KNOWN AS THE GAMMA FACTOR.

g
1
1

V
2
C
2
g
1
1

V
2
C
2
g
1
1

V
2
C
2
g
1
1

V
2
C
2
Denominator
g
1
1

V
2
C
2
Denominator
Another
fraction, with the
numerator and
denominator
squared
g
1
1

V
2
C
2
Denominator
Another
fraction, with the
numerator and
denominator
squared
g
v
1
1
V
2
C
2
is the speed at which
the mass is moving
c
(as you already know)
is the speed of light

2
v
Also, notice that we can write
c2
as



v  2
c 
Under the square root (in the denominator), we have
2
v
1 
c



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.
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: 0.000000073]
If we now square that answer, we get:
0.0000000000000054
Finally,
1 - 0.0000000000000054 = 0.999999999999946
which, for most purposes, can be written simply as:
1
So, for “everyday speeds”,
1
v2
≈ 1
c2
AND
 g 
1
1


V
2
C
2
≈
1
1
=
1
NOW,
as v gets larger, so does
v2
c2
Thus, 1 
v2
will become smaller,
c2
and g 


1
1
V
2
C
2
will become larger.
To prove to yourself that this is true,
take a speed v of 99.5% the speed of light (0.995c):
v  2
Then, c 



= (0.995)2 ≈ 0.99
and 1 
v2
≈ 0.01
c2
So, g 

1
1

V
2
C
2
≈
1
= 10
0.01
AND WHAT HAPPENS WHEN V = C ?
Then 1 
v2
=0
c2
AND

1
g
0
=
1
0
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.
THIS GRAPH SHOWS HOW THE GAMMA FACTOR VARIES WITH SPEED:
g-factor vs. speed
g
0.1c
0.2c
0.3c
0.4c
0.5c
0.6c
0.7c
0.8c
0.9c
1.0c
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.
Speed
(v)
0.00000009259c
g
(to 3 decimal places)
1.000
(= 100 km/h)
0.1c
0.2c
0.3c
0.4c
0.5c
0.6c
0.7c
0.8c
0.9c
0.95c
0.99c
0.995c
0.999c
0.9999c
0.99999c
1.005
1.021
1.048
1.091
1.155
1.250
1.400
1.667
2.294
3.203
7.089
10.013
22.366
70.712
233.607
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 = gmc2
ACTUALLY TELL US?
SO MUCH FOR THE GAMMA FACTOR, BUT WHAT DOES THE FORMULA
E = gmc2
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)
SO MUCH FOR THE GAMMA FACTOR, BUT WHAT DOES THE FORMULA
E = gmc2
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 = gmc2, is actually the sum of these two:
Total energy = rest energy + kinetic energy
OR
gmc2 = mc2 + Ek
Also note that, if we are interested in finding only the kinetic energy,
we can rearrange gmc2 = mc2 + Ek and write
Ek = gmc2 − mc2
OR
Ek = (g − 1)mc2
Note that for a speed of 0.995c, Ek ≈ (10 − 1)mc2 = 9mc2.
This can sometimes be useful for rough approximations.
#3
ig 



m   0
ig 




m   0
ig 




m   0
ig 




m   0
NO!!!!
NO!!!!
NO!!!!
NO!!!!
NO!!!!
NO!!!!
NO!!!!
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.
It is called the Dirac Equation, named for the physicist
who formulated it, Paul A. M. Dirac.
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.
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,
16 = +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.

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:
http://cerncourier.com/cws/article/cern/28693