Introduction
One of the most extraordinary things about Einstein’s energymass
equivalence equation
is its simplicity. However, we still need to make sure we are using the correct
units when solving the equation, and that we understand the answer. The purpose
of this page is to solve the equation as it is and give some idea of the huge
amount of energy locked up in even the smallest amount of mass.
The Components of the Equation
If we break the equation E = mc^{2} into its components and write out
the terms fully we get:
 E = energy (measured in joules)
 m = mass (measured in kilograms)
 c = the speed of light (186,000 miles per second, or 3
x
10^{8} ms^{1})
We will now examine each of the terms in a little more detail.
Energy
Energy is measured in joules (J). How much energy is one joule? Not very
much really. If you pick up a large apple and raise it above your head you will
have used around one joule of energy in the process. On the other hand, we use up
huge amounts of energy every time we switch on a
light. A 100 watt light bulb
uses 100 joules of energy every second, i.e. one watt is one joule per
second.
Mass
Mass is a measure of a body’s resistance to acceleration. The greater the
mass the greater the resistance to acceleration, as anyone who has ever tried
to push a heavy object knows. For our purposes here, we can also think of mass
as the amount of matter in an object. Mass is measured in
kilograms (kg),
with 1kg
about the same as 2.2 pounds. Note that we haven’t said what the mass is
composed of. In fact, it could be anything. It doesn’t matter if we use
iron, plastic, wood, rock or gravy. The equation tells us that
whatever the mass is it can be turned into energy (whether it is practical to
actually do so is another matter and is dealt with in other pages in this
series).
The speed of light
The speed of light is very close to 186,300 miles per second (300,000 km per
second). In order to make the equation "work" we need to convert these
numbers into units that are more suited to our purposes. In physics, speeds are
measured in metres per
second. This is usually abbreviated to ms^{1};
that is "metres times seconds to the minus one". Don’t worry if you
don’t understand this notation. We could equally write m/s but using ms^{1}
makes the mathematics easier in the long run. Likewise, we could either say that
the speed of light is 300,000,000 metres per second, or, as is more usual,
express the
same figure in scientific notation: 3 x 10^{8}
ms^{1}.
Solving the Basic Equation
Now that we have everything in order let’s have a go at solving the
equation. We will use a mass of 1kg to keep things simple and I will show all of
the workings of the equation. So, with 1kg of mass (around 2.2 pounds) we get:
Note how the units were dealt with and that kg m^{2} s^{2}
is the same as joules (although a rigorous proof of this is outside the scope of these
pages).
So from 1kg of matter, any matter, we get 9
x
10^{16} joules of energy. Writing that out fully we get:
90,000,000,000,000,000 joules
That is a lot of energy! For example, if we converted 1kg of mass
into energy and used it all to power a 100 watt light bulb how long could we
keep it lit for? In order to answer the question the first thing to do is divide the result by watts (remember
that 1 watt is 1 joule per second):
9 x 10^{16} J /
100W = 9 x 10^{14} seconds
How long is that in years? A year (365.25 days) is 31,557,600 seconds, so we
get:
9 x 10^{14} seconds
/ 31,557,600 seconds = 28,519,279 years
That is a very long time!
Of course, converting mass into energy is
not quite that simple, and apart from with some tiny particles in experimental
situations has never been carried out with 100% efficiency. Perhaps that’s just
as well.
Conclusion
We have seen that the E = mc^{2} equation is easy to solve as it is and that for
even a small amount of mass a huge amount of energy can, at least in theory, be
released. Other pages in this
series show how the energy can be
released in practical ways, as well as deriving the equation in both
simple and
complex terms.
E = mc^{2}
A huge amount
of energy from a small amount of mass
