A5-5 Special Relativity: Mass-Energy EquivalenceHL ONLY
1. The Equivalence of Mass and Energy
One of the most profound consequences of Einstein's Special Relativity is the realization that mass and energy are not two separate entities, but rather two forms of the exact same thing. Mass can be converted into pure energy, and energy can be converted into mass.
Mass-Energy Equivalence Formula:
- $\Delta E$: The change in energy (Joules, J)
- $\Delta m$: The change in mass, also known as the mass defect (kg)
- $c$: The speed of light in a vacuum ($3.00 \times 10^8\text{ m/s}$)
The implications of $c^2$: Because the speed of light squared ($c^2 \approx 9 \times 10^{16}$) is such an astronomically large number, this formula tells us that a tiny, almost imperceptible amount of mass is equivalent to an absolutely massive amount of energy. This is the foundational principle behind both nuclear power and nuclear weapons.
2. Applications in Particle Physics
In classical physics, we are taught the "Law of Conservation of Mass" and the "Law of Conservation of Energy." In relativistic physics, these merge into a single law: the Conservation of Mass-Energy.
- Nuclear Fission & Fusion: When atomic nuclei split (fission) or combine (fusion), the total mass of the resulting products is slightly less than the mass of the starting materials. This "missing mass" ($\Delta m$) has not vanished; it has been released as kinetic energy and radiation ($\Delta E$).
- Particle Annihilation: When a particle meets its antimatter equivalent (e.g., an electron colliding with a positron), 100% of their combined mass is entirely converted into pure energy (photons), leaving no mass behind.
- Pair Production: The reverse is also true. A high-energy photon can spontaneously convert its pure energy into mass, creating an electron-positron pair out of seemingly nothing.
Example 4: Energy of a Grain of Sand
Problem: Suppose you could entirely convert a single grain of sand, with a mass of just $1.0 \times 10^{-6}\text{ kg}$ (one milligram), into pure energy. How much energy would be released?
Solution:
- Step 1: Identify the given variables.
$\Delta m = 1.0 \times 10^{-6}\text{ kg}$
$c = 3.00 \times 10^8\text{ m/s}$ - Step 2: Apply the mass-energy equivalence formula.
$$ \begin{aligned} \Delta E &= \Delta m c^2 = (1.0 \times 10^{-6}) \times (3.00 \times 10^8)^2 = (1.0 \times 10^{-6}) \times (9.00 \times 10^{16}) \\ &= \mathbf{9.0 \times 10^{10}\text{ J}} \end{aligned} $$ - Perspective: 90 billion Joules is roughly the same amount of energy released by detonating 21 tons of TNT—all from a single speck of sand!