The equation $E=mc^2$ is Albert Einstein's famous mass-energy equivalence principle, which states that mass ($m$) and energy ($E$) are different manifestations of the same physical entity and can be converted into one another. The $c^2$ term is the speed of light squared, acting as a massive conversion factor.
While the full derivation from the principles of Special Relativity is mathematically complex, a simplified conceptual derivation can be performed by considering a body moving near the speed of light, where the concept of relativistic mass is introduced.
⚡️ Simplified Conceptual Derivation
One common way to conceptually illustrate this relationship is to consider an object moving at a velocity close to the speed of light ($c$), and a constant force ($F$) acting on it for a unit of time ($\Delta t=1$).
1. Energy Gained
The energy ($E$) gained by the body is equal to the work done on it by the force. Work is defined as force times the distance traveled ($\text{distance} = \text{velocity} \times \text{time}$).
$E = F \times \text{distance}$
Since the body's velocity is nearly $c$, the distance traveled in a unit time ($\Delta t=1$) is $c$.
This gives the first key relationship:
$$E = F \times c \quad \text{(Eq. 1)}$$
2. Force and Momentum
From Newton's laws, force ($F$) is the rate of change of momentum ($p$) over time, or $F = \Delta p / \Delta t$. Momentum is defined as mass times velocity ($p = m \times v$).
In this relativistic scenario, since the body is already moving near $c$, the force cannot significantly change the velocity ($v \approx c$). Instead, the force primarily increases the body's mass ($m$).
The change in momentum ($\Delta p$) is therefore due to the change in mass ($\Delta m$): $\Delta p = \Delta m \times c$.
Since we are considering a unit time ($\Delta t=1$), the force is equal to the change in momentum: $F = \Delta p / 1$.
This gives the second key relationship:
$$F = \Delta m \times c \quad \text{(Eq. 2)}$$
3. Combining the Equations
Now, substitute Equation 2 into Equation 1:
Simplifying this yields:
This result demonstrates that the change in energy ($\Delta E$) is directly proportional to the change in mass ($\Delta m$) by the factor of $c^2$. If the energy $E$ is considered the rest energy of a body with mass $m$, the equation becomes the final famous form:
This is the special case where $E$ is the energy contained in an object at rest (its rest energy), and $m$ is its rest mass. The more general relativistic energy-momentum relation for a moving particle with momentum $p$ and rest mass $m_0$ is:
If the particle is at rest ($p=0$), this reduces to $E^2 = (m_0 c^2)^2$, or $E = m_0 c^2$.
This video offers a two-minute visualization of the derivation of $E=mc^2$:
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