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 =

mc2

#3

i m 0

#1

E = hf

Hello.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.

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

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 × 1034 joule seconds (Js)

WE CAN ALSO EXPRESS IT IN DIFFERENT UNITS:

h = 4.14 × 1015 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 × 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.

E hc

f

INCREASING ENERGYINCREASING 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 = mc2

As the CEO of a large corporation, this

formula seems quite difficult to me, and

has begun to make my head hurt a lot.

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:

CAN BE USED TO CONVERT ENERGY TO MASS.

mEc2

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 MOVINGWE NEED TO CHANGE THIS FORMULA A LITTLE…

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)

1

1 V2

C2

AAAAHHHHHH!

AAAAHHHHHH!

1

1 V2

C2

AAAAHHHHHH!

AAAAHHHHHH!

THIS EXPRESSION IS KNOWN AS THE GAMMA FACTOR.

1

1 V2

C2

IT’S

JUST A

FRACTION

—

1

1 V2

C2

IT’S

JUST A

FRACTION

—

GONE MAD!

1

1 V2

C2

IT’S

JUST A

FRACTION

—

GONE MAD!

Numerato

r

1

1 V2

C2

Numerato

r

Denominator

IT’S

JUST A

FRACTION

—

GONE MAD!

1

1 V2

C2

Another fraction, with

the numerator

and denominator

squared

IT’S

JUST A

FRACTION

—

GONE MAD!

Numerato

r

Denominator

1

1 V2

C2

1

1 V2

C2

Square ro

ot

of the

denomin

ator

IT’S

JUST A

FRACTION

—

GONE MAD!

Another fraction, with

the numerator

and denominator

squared

Numerato

r

Denominator

v is the speed at

which the mass is

moving

c (as you already

know) is the speed of

light

v2

c2

Also, notice that we can write as 2

cv

1

1 V2

C2

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.

2

cv1

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 ×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

AND

1v2

c2

≈

11

= 1

1

1 V2

C2

NOW,

as v gets larger, so does

Thus, will become smaller,

and will become larger.

v2

c2

1v2

c2

1

1 V2

C2

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

2

cv

1v2

c2

10.01

1

1 V2

C2

AND WHAT HAPPENS WHEN V = C ?

Then = 0

AND

=

WHICH IS NOT ALLOWED IN MATHEMATICS.

1v2

c2

10

10

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.

0.1c 0.2c 0.5c0.3c 0.6c0.4c 1.0c0.7c 0.9c0.8c

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 FINDTHE GAMMA FACTOR

FOR VARIOUS SPEEDS

SO MUCH FOR THE GAMMA FACTOR, BUT WHAT DOES THE FORMULA

E = mc2

ACTUALLY TELL US?

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)

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

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.

i m 0

i m 0

i m 0

i m 0

NO!!!!

NO!!!!

NO!!!!

NO!!!!

NO!!!!

NO!!!!

NO!!!!

NO!!!!

NO!!!!

NO!!!!

NO!!!!

NO!!!!

NO!!!!

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.

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.

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…

16

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