Without doubt E = mc 2 is the world’s most famous equation. This page explains what E = mc 2 means in simple terms and some of its consequences. The equation is derived directly from Einstein’s Special Theory of Relativity, and other pages in this series deal with the mathematical and logical derivation. Here though, we will examine the equation as it stands and keep the mathematics to a minimum.

Energy = mass x the speed of light squared

In other words: E = energy (measured in joules, J) m = mass (measured in kilograms, kg) c = the speed of light (measured in meters per second, ms -1 ), but this needs to be "squared". Note that the case of each letter is important and it would be incorrect to show the equation as, for example, e = MC². This is because physicists use the case of letters as well as the letters themselves to denote particular physical entities, quantities and constants in equations. In order for the equation to be correct we need to "square" the term c (the speed of light), i.e. we multiply the speed of light by itself; hence c 2 is the same as c times c. This allows us to be write the equation in another, slightly unusual, but equally correct way: E = m x c x c As a matter of interest, and to complete the terms used in the equation, the equals sign was only invented during the 16th century, by the Welsh mathematician Robert Recorde, apparently unhappy having to write out "is equal to" in his work. He could have chosen any number of symbols but chose two parallel lines because, as he himself put it, "noe 2 thynges can be moare equalle". We will now examine each unit (i.e. letter) in the equation in turn before addressing the question of what the equation means, but if you want to see worked examples of the equations you can do so here.

Introduction

The word "energy" is actually quite new. Its modern use dates from around the middle of the nineteenth century, when it was beginning to be realised that the power that drove many different processes could be explained by the concept of energy being transferred from one system and form to another. For example, the trains of the day were powered by coal. The coal was burned under a water-filled boiler to produce steam, which in turn pushed pistons attached to the wheels of the train, the wheels turned and the train was set in motion. In this example we start with locked up ("latent") chemical energy in the coal. The chemical energy is turned into heat energy (sometimes called "thermal energy") by burning the coal and boiling the water. Finally, the thermal energy is turned into the energy of movement ("kinetic energy") by forcing the steam into pistons to drive the wheels:

A moving steam train Chemical energy - thermal energy - kinetic energy

E = Energy

There are many other forms of energy, such as electrical, gravitational, nuclear, and strain energy, such as that found in springs. However, as different as all these types of energy seem they can all be measured in the same way and thought of as the same thing. The unit that we use to measure energy, from whatever energy source, is the joule (J). Two ways in which we use this unit in everyday terms are:

The total amount of energy in a system:

As noted above, one example is a lump of coal, which when burned will release a certain number of joules (J) of energy, mostly in the forms of heat and light. Another, perhaps more common example, is that it takes about 1 joule to raise an apple by 1 meter.

Energy used up over time:

Most electrical devices have their power consumption rated in watts (W). A watt is a rate of energy consumption of one joule per second. So, if you have a light bulb in your room that's rated at 100 W it's using energy at a rate of 100 joules every second.

To go back to the second example in the first bullet point, lifting an apple by 1 metre every second would mean that there is a power output of 1 watt. For most people this would be quite easy and could be kept up for quite a long time, but now imagine lifting 100 apples a second, i.e. 100 W. This, in human terms, is a large power output, but nothing special for many electrical devices. It's not uncommon, for example, for a kettle to be rated at 2000 W or more. That's a lot of apples!

So, to summarise, energy comes in many forms, and it can be transferred from one system to another. The basic unit of measurement for energy is the joule.

m = Mass

c = the Speed of Light

Note that mass isn’t the same as weight, although it's often thought to be. Weight is actually a measure of the gravitational force (pull) felt by a body and is measured in newtons (N) (note that scientific units that are named after people are almost always in lower case when spelled out fully, hence newtons and not Newtons, watts and not Watts etc.). For example, astronauts walking on the surface of the Moon have the same mass as on Earth but only weigh one sixth of what they would do back home. The reason for this is that while the mass of the astronauts hasn’t changed, the pull of the Moon’s gravity is only one sixth of what the Earth’s gravitational pull is.

As with energy, the idea that mass is common to all objects is relatively new and again dates back to around the nineteenth century. Before that time different solids, liquids and gases were all thought to be only loosely connected in conceptual terms. As with energy, we now consider that mass is neither created or destroyed, but is merely changed from one form to another, e.g. we can turn water from a solid (ice) into a liquid (water) and into a gas (steam), but its total mass doesn’t change.

The speed of light was first accurately estimated by the Danish astronomer Ole Roemer (sometimes written as Rømer) during the 1670s. Up until that time everyone assumed that the speed of light was infinite, i.e. that light arrived at its destination instantly. This isn’t such an unreasonable assumption given that when we look around us light does indeed appear to reach us instantly.

However, during the seventeenth century it was discovered that there was a problem in calculating the orbital time of Io, the innermost moon of Jupiter. It sometimes took "too long" to make an orbit of the planet and at other times was "too quick". It was thought that the problem must be due to a wobble in the orbit of Io, but Roemer took a different, and very radical, view of the matter. He argued that light, instead of being everywhere instantly, had a finite speed and that this would explain the problem of Io. The Earth was known to travel around the Sun and this meant that sometimes the Earth was closer to Jupiter and sometimes further away. Roemer realised that when the Earth was on the opposite side of the Sun from Jupiter the light from Io would take longer to reach us than when the two bodies were on the same side:

You may think that that was the end of the matter and that Roemer was celebrated as a scientific genius, showered with honours and given a secure future. Sadly, that’s far from what happened. He was only 21 when he made his discovery, while Cassini was a well-respected if egotistical elder scientist, who used his powerful friends to back him up to rubbish Roemer’s ideas. Scientists, it seems, are human after all and this wasn’t the first, or sadly, the last time that an ego got in the way of a new discovery. Roemer eventually gave up science completely and later became the director of the port of Copenhagen and then head of the State Council of the Realm. It wasn’t until 50 years later that further experiments convinced the scientific community that Roemer had been right all along.

Ole Roemer 1644 - 1710

What Does the Equation Mean?

The equation tells us that energy and mass are, effectively, the same thing, and it also tells us how much energy is contained in a given mass, or vice versa. In other words, mass can be thought of as very tightly packed energy. That energy and mass are equivalent is quite an extraordinary claim and seems to go against two laws that had been established by scientists before Einstein came along:

The Law of the Conservation of Mass:

As we have seen, mass can be thought of as the quantity of matter in an object. The law of the conservation of mass states that mass is always conserved. That is, whatever we do with matter in a closed system we will always have the same amount of substance at the end. For example, if we burn a log, the wood gets lighter as the fuel it contains is used up. However, if we gather together the ashes, all of the tiny smoke particles and the water vapour produced by the burning process and then weigh everything we find that the mass is exactly equal to the mass of the log that was burned. Mass is just mass, or so it seems, and while it can be chemically altered, such as burned, the total amount in any system remains the same.

The Law of the Conservation of Energy:

But what about the energy released in burning the log? The energy released in the burning process is "chemical energy", i.e. the breaking and reforming of chemical bonds between atoms and molecules. Burning the wood released the chemical energy locked up in it. No energy was created in the process and none was destroyed; it was just changed from one sort of energy (chemical bonds) to other forms of energy (heat and light). In other words the total amount of energy, just like the total amount of mass, remained the same.

What Einstein showed via his now famous equation was that mass and energy are in fact the same thing. Converting one into the other doesn’t therefore violate either of the two conservation laws. Both quantities are conserved, although the state of the mass/energy may have changed. Each atom of a substance can be thought of as a little ball of tightly packed energy that can be released under certain circumstances. Likewise, we can take energy (such as particles of light, called photons) and turn it into matter. This was first achieved in the 1930s.

That light can be turned into matter is perhaps a rather odd idea, but the picture below shows the first successful experiment in which this was done:

Cloud chamber photon decay

The picture shows the tracks of two matter particles that have been "created" after a high energy photon decayed, i.e. "fell apart", in a cloud chamber. The high energy photon is not in the visible range and has entered the chamber from the bottom of the picture.

The Cloud Chamber

A cloud chamber is a sealed tank filled with a gas, usually with a magnet to one side of it. When a particle, such as an atom, electron or proton passes through the tank it collides with some of the particles in the gas to produce little clouds that mark its path. For an electrically neutral particle, such as a neutron, the path will be straight. However, for any particle that is not electrically neutral its path will be bent towards or away from the magnet that forms part of the apparatus.

The subject of turning matter into energy through both fusion and fission is dealt with in other pages in this series.

Einstein’s Explanation of his Equation

The recording is old, and that, together with Einstein’s accent, sometimes makes it difficult to hear the words properly. This is a transcript of the recording:

“It followed from the Special Theory of Relativity that mass and energy are both but different manifestations of the same thing - a somewhat unfamiliar conception for the average mind. Furthermore, the equation E is equal to mc 2 , in which energy is put equal to mass, multiplied with the [by the] square of the velocity of light, showed that very small amounts of mass may be converted into a very large amount of energy and vice versa. The mass and energy were in fact equivalent, according to the formula mentioned before [E = mc 2 ]. This was demonstrated by Cockcroft and Walton in 1932, experimentally."

Albert Einstein (1879 - 1955)

Albert Einstein (1879 - 1955)

What do the Letters Stand For?

Each of the letters of E = mc 2 stands for a particular physical quantity. Writing them out in full we get:

E = mc2 The Basics

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Energy = mass x the speed of light squared

Introduction

A moving steam train Chemical energy - thermal energy - kinetic energy

E = Energy

m = Mass

c = the Speed of Light

Ole Roemer 1644 - 1710

What Does the Equation Mean?

Cloud chamber photon decay

The Cloud Chamber

The subject of turning matter into energy through both fusion and fission is dealt with in other pages in this series.

Einstein’s Explanation of his Equation

Albert Einstein (1879-1955)

What do the Letters Stand For?

Each of the letters of E = mc 2 stands for a particular physical quantity. Writing them out in full we get:

In other words:

E = energy (measured in joules, J) m = mass (measured in kilograms, kg) c = the speed of light (measured in meters per second, ms -1 ), but this needs to be "squared".

Note that the case of each letter is important and it would be incorrect to show the equation as, for example, e = MC 2 . This is because physicists use the case of letters as well as the letters themselves to denote particular physical entities, quantities and constants in equations.

In order for the equation to be correct we need to "square" the term c (the speed of light), i.e. we multiply the speed of light by itself; hence c² is the same as c times c. This allows us to be write the equation in another, slightly unusual, but equally correct way:

E = m x c x c

As a matter of interest, and to complete the terms used in the equation, the equals sign was only invented during the 16th century, by the Welsh mathematician Robert Recorde. Apparently unhappy having to write out "is equal to" in his work. He could have chosen any number of symbols but chose two parallel lines because, as he himself put it, "noe 2 thynges can be moare equalle".

We will now examine each unit (i.e. letter) in the equation in turn before addressing the question of what the equation means, but if you want to see worked examples of the equations you can do so here.

The total amount of energy in a system:

Energy used up over time:

So, to summarise, energy comes in many forms, and it can be transferred from one system to another. The basic unit of measurement for energy is the joule.

Mass is strictly defined as a measure of a body’s inertia, i.e. its resistance to acceleration. Another and simpler way of defining mass is to say that it's the total amount of matter in an object. This latter definition isn’t strictly true, but is good enough for our purposes here. Mass is measured in kilograms (kg). Note that mass isn’t the same as weight, although it's often thought to be. Weight is actually a measure of the gravitational force (pull) felt by a body and is measured in newtons (N) (note that scientific units that are named after people are almost always in lower case when spelled out fully, hence newtons and not Newtons, watts and not Watts etc.). For example, astronauts walking on the surface of the Moon have the same mass as on Earth but only weigh one sixth of what they would do back home. The reason for this is that while the mass of the astronauts hasn’t changed, the pull of the Moon’s gravity is only one sixth of what the Earth’s gravitational pull is. As with energy, the idea that mass is common to all objects is relatively new and again dates back to around the nineteenth century. Before that time different solids, liquids and gases were all thought to be only loosely connected in conceptual terms. As with energy, we now consider that mass is neither created or destroyed, but is merely changed from one form to another, e.g. we can turn water from a solid (ice) into a liquid (water) and into a gas (steam), but its total mass doesn’t change.

We use the letter c to represent the speed of light. The ‘c’ comes from the Latin word “celeritas”, meaning swift, and it’s a very apt definition - there is nothing faster than light. In a vacuum, such as space, it travels at close to 186,300 miles per second (300,000 km per second). That’s about seven times around the Earth every second. The speed of light was first accurately estimated by the Danish astronomer Ole Roemer (sometimes written as Rømer) during the 1670s. Up until that time everyone assumed that the speed of light was infinite, i.e. that light arrived at its destination instantly. This isn’t such an unreasonable assumption given that when we look around us light does indeed appear to reach us instantly. However, during the seventeenth century it was discovered that there was a problem in calculating the orbital time of Io, the innermost moon of Jupiter. It sometimes took "too long" to make an orbit of the planet and at other times was "too quick". It was thought that the problem must be due to a wobble in the orbit of Io, but Roemer took a different, and very radical, view of the matter. He argued that light, instead of being everywhere instantly, had a finite speed and that this would explain the problem of Io. The Earth was known to travel around the Sun and this meant that sometimes the Earth was closer to Jupiter and sometimes further away. Roemer realised that when the Earth was on the opposite side of the Sun from Jupiter the light from Io would take longer to reach us than when the two bodies were on the same side:

This means that the light has to travel further and therefore takes longer, providing, of course, that light has a speed in the first place. During a meeting of the new Academy of Science in Paris in 1676 Roemer demonstrated that the amassed observational data of the astronomer Cassini indicated that Io would next appear at 5.25pm on 9th November of that year. He himself predicted that it wouldn’t appear until 10 minutes and 45 second later, using his theory that light had a finite speed. The day came and virtually every major observatory in Europe was ready to test the prediction. At 5.25pm, the time predicted by Cassini, Io wasn’t visible. Even at 5.35pm Io wasn’t visible. But at exactly 5.35pm and 45 seconds it appeared, just as Roemer said it would. From this it was possible to make the first accurate measurement of the speed of light and the calculated figure was within one percent of what we know it to be today. You may think that that was the end of the matter and that Roemer was celebrated as a scientific genius, showered with honours and given a secure future. Sadly, that’s far from what happened. He was only 21 when he made his discovery, while Cassini was a well-respected if egotistical elder scientist, who used his powerful friends to back him up to rubbish Roemer’s ideas. Scientists, it seems, are human after all and this wasn’t the first, or sadly, the last time that an ego got in the way of a new discovery. Roemer eventually gave up science completely and later became the director of the port of Copenhagen and then head of the State Council of the Realm. It wasn’t until 50 years later that further experiments convinced the scientific community that Roemer had been right all along.

After many experiments, notably by the scientist for whom the unit of energy is named, James Prescott Joule (1818 - 1889), it was established that the total amount of energy in a closed system always remains the same. This is known as the law of the conservation of energy. What Einstein showed via his now famous equation was that mass and energy are in fact the same thing. Converting one into the other doesn’t therefore violate either of the two conservation laws. Both quantities are conserved, although the state of the mass/energy may have changed. Each atom of a substance can be thought of as a little ball of tightly packed energy that can be released under certain circumstances. Likewise, we can take energy (such as particles of light, called photons) and turn it into matter. This was first achieved in the 1930s. That light can be turned into matter is perhaps a rather odd idea, but the picture below shows the first successful experiment in which this was done:

Einstein must have spent a great deal of his life explaining his most famous equation. Thankfully, though, someone recorded him explaining it and you can hear it in his own words by downloading the file here: The recording is old, and that, together with Einstein’s accent, sometimes makes it difficult to hear the words properly. This is a transcript of the recording:

The Law of the Conservation of Mass: The Law of the Conservation of Energy: As we have seen, mass can be thought of as the quantity of matter in an object. The law of the conservation of mass states that mass is always conserved. That is, whatever we do with matter in a closed system we will always have the same amount of substance at the end. For example, if we burn a log, the wood gets lighter as the fuel it contains is used up. However, if we gather together the ashes, all of the tiny smoke particles and the water vapour produced by the burning process and then weigh everything we find that the mass is exactly equal to the mass of the log that was burned. Mass is just mass, or so it seems, and while it can be chemically altered, such as burned, the total amount in any system remains the same. But what about the energy released in burning the log? The energy released in the burning process is "chemical energy", i.e. the breaking and reforming of chemical bonds between atoms and molecules. Burning the wood released the chemical energy locked up in it. No energy was created in the process and none was destroyed; it was just changed from one sort of energy (chemical bonds) to other forms of energy (heat and light). In other words the total amount of energy, just like the total amount of mass, remained the same.

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