Mass and energy are related quantities that we can respectively convert to the other by using the speed of light. In physics, mass is considered to be a form of energy because the mass of a particle can be converted into other forms of energy such as thermal energy, kinetic energy, etc. In the same way, kinetic energy or radiant energy can form particles with mass.
Mass-energy conversion
There are many examples of mass-energy conversions that occur in daily life. One of them is the reason the universe exists, the Big Bang. According to the Big Bang theory, the universe was formed after a gigantic amount of energy was released and converted into mass.
Other great examples of energy-mass conversion include nuclear power plants, nuclear weapons, and the sun. However, there is a difference in the way they convert mass to energy. For instance, while the sun converts mass to energy via nuclear fusion, nuclear power plants do it via nuclear fission.
How can mass be converted into energy?
The first postulate of Einstein’s theory of special relativity states that the laws of physics are the same for all inertial frames of reference. When properly studied, this theory yields an expression for relativistic energy whose dependence on the inertial frame of reference is captured by a relativistic factor.
In order to understand relativistic energy, we have to take into consideration total energy and rest energy.
Total energy
Total energy E is the sum of all energies that an object with mass carries. Mathematically, it can be defined as:
Here, m is the mass, c is the speed of light, and \(\gamma\) can be calculated as follows:
\[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\]
Here again, v is the velocity of the moving object [m/s], while c is the speed of light in a vacuum [m/s].
It’s clear that E is related to relativistic momentum, which is the momentum of an object travelling at a relativistic speed. However, if the velocity is zero, \(\gamma\) will not be zero. It rather tells us that the object has some rest energy.
Rest energy
Rest energy can be defined as the energy of an object whose velocity with respect to the frame of reference is zero. It can be described mathematically as:
\[E_0 = mc^2\]
E0 is the rest energy in joules.
This might remind you of Einstein’s most famous equation, which showed that when an object is at rest, its energy is proportional to its mass. Thus, when energy is stored in an object, its rest mass increases. This also shows that destroying mass can release energy. See the following example for further illustration.
Calculate the rest energy of a proton in joules.
The mass of a single proton is approximately \(1.67 \cdot 10^{-27}\) kg, and we know that the speed of light c is \(3.00 \cdot 10^8 m/s\). Let’s put these values into the following equation:
\[E_0 = mc^2\]
This gives us:
\[E_0 = (1.67 \cdot 10^{-27})\cdot (3\cdot 10^8)^2 = 1.503\cdot 10^{-10}[kg \cdot m^2/s^2]\]
Let’s now convert the unit to joules to see how large the energy is. We know that:
\[1 [kg \cdot m^2/s^2] = 1 [Joule]\]
The result, therefore, is:
\[E_0 = 1.503 \cdot 10^{-10} [J]\]
This may seem small, but we need to keep in mind that this is only for a single proton. If you wanted to calculate the rest energy of 1 gram mass, you might be surprised that the result would be \(9 \cdot 10 ^ {13} J\). We don’t notice this because usually, the energy is not available.
How can energy be converted into mass?
Just as mass can be converted into energy, energy can also be converted into mass. In order to understand how this works, we need to look into the relationship between stored energy and potential energy.
Stored energy and potential energy
What happens to the energy that is stored in a compressed spring? It becomes part not only of the total energy but also of the mass of the spring. So, how come we don’t notice these changes in mass? Let’s look at the example below.
Let’s say a big battery is able to move 700 ampere-hours (A ⋅ h) of charge at 15 volts. Calculate the difference in mass when the battery goes from fully discharged to fully charged.
First, using the equation below, let’s find out how much energy can be stored in the battery in the form of electrical potential (PE) energy:
\[PE_{Electrical} = q \cdot V\]
Here, q is the charge in ampere-hours, while V is the electrical potential difference in Volts.
The change of energy is in the form of electrical potential energy, and we want to determine the difference in mass. So, let’s put together the previous equation and the one for finding the rest energy:
\[\Delta E = PE_{Electrical} = q \cdot V = (\Delta m) \cdot c^2\]
The charge q, we said, is 700 A⋅h, while V is 15 V. This gives us:
\[\Delta m = \frac{q \cdot V}{c^2}\]
\[\Delta m = \frac{(700 [A \cdot h])\cdot (15 [V])}{(3.00 \cdot 10^8)^2}\]
Next, we write amperes as coulombs per second and convert hours into seconds.
\[\Delta m = \frac{(700 [C/s]\cdot (3600 [s] \cdot 15 [J/C]))}{(3.00 \cdot 10^8)^2}\]
\[\Delta m = \frac{(2.52 \cdot 10^6) \cdot (15 [J/C])}{(3.00 \cdot 10^8)^2}\]
Knowing that \(1kg \cdot (\frac{m ^ 2}{ s ^ 2}) = 1 \space Joule\), we can change the result to kilograms, which gives us the difference in mass as follows:
\[\Delta m = 4.2 \cdot 10^{-10} [kg]\]
As you can see, there is only a very small amount of change in the mass, which explains why we don’t notice this phenomenon in our daily lives.
Mass and Energy - Key takeaways
- Mass and energy are the same quantities, which we can convert into each other by using the speed of light.
- Total energy E is the sum of all energies that an object with mass carries. Mathematically, it can be defined as \(E = \gamma mc ^ 2\).
- Rest energy can be defined as the energy of an object whose velocity is zero. It can be stated mathematically as \(E_0 = mc ^ 2\).
- Just as mass can be converted into energy, energy can also be converted into mass. However, in everyday contexts, the change in mass is so small that we don’t notice it.
- The change in mass when there is excess energy stored can be determined with the equation \(E = PE = q \cdot V = (\Delta m) \cdot c ^ 2\).
- The forming of the universe (Big Bang), nuclear power plants, and the sun are all examples of mass-energy conversion.
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