Mass-energy equivalence and Bertozzi's experiment

Relativistic mass and rest energy

• In special relativity, the mass of a particle increases with speed. The relativistic massThe effective mass of a particle at speed $v$, given by $m = \gamma m_0$, where $m_0$ is the rest mass. As $v$ approaches $c$, the relativistic mass tends to infinity. is $m = \gamma m_0$, where $m_0$ is the rest mass.
• As $v \to c$, the Lorentz factor $\gamma \to \infty$, so the relativistic mass tends to infinity. This is why no particle with mass can ever reach the speed of light: it would require infinite energy.
• Einstein showed that energy and mass are equivalent:

• The total energy of a particle is the sum of its rest energyThe energy a particle has purely by virtue of its mass, when at rest. Given by $E_0 = m_0 c^2$. and its kinetic energy: $E_{\text{total}} = m_0 c^2 + E_k$

Relativistic kinetic energy

• The relativistic kinetic energy is found by subtracting the rest energy from the total energy:

• At low speeds ($v \ll c$), $\gamma \approx 1 + \frac{v^2}{2c^2}$, so $E_k \approx \frac{1}{2}m_0 v^2$, recovering the Newtonian expression.
• At high speeds, the relativistic kinetic energy is significantly larger than $\frac{1}{2}m_0 v^2$. The difference grows dramatically as $v$ approaches $c$.

Bertozzi's experiment (1964)

• William Bertozzi accelerated electrons to speeds close to $c$ and directly measured their kinetic energies to test whether they followed Newtonian or relativistic predictions.
• Method: Electrons were accelerated through a known potential difference $V$, giving them kinetic energy $E_k = eV$. They travelled through an evacuated tube and hit an aluminium target.
• The speed was measured using the time of flight: a signal was triggered when the electrons left the electric field, and another when they hit the target. The distance between the two detectors was known.
• The results were plotted as $v^2/c^2$ against $E_k/(m_e c^2)$.

The results

• The Newtonian prediction ($E_k = \frac{1}{2}m_0 v^2$, so $v^2/c^2$ is proportional to $E_k$) gives a straight line through the origin with no upper limit on speed.
• The relativistic prediction gives a curve that approaches $v^2/c^2 = 1$ asymptotically. The speed never reaches $c$, no matter how much energy is added.
• Bertozzi's experimental data points fell on the relativistic curve, not the Newtonian line. This was direct confirmation that no particle with mass can reach the speed of light.

Verifying the kinetic energy

• To confirm that $E_k = eV$ was valid, Bertozzi measured the kinetic energy independently by recording the temperature rise of the aluminium target.
• Using $\Delta E = mc\Delta\theta$ (where $m$ is the target mass, $c$ is the specific heat capacity of aluminium, and $\Delta\theta$ is the temperature rise), the total energy deposited was found.
• Dividing by the number of electrons (from the total charge $Q$ hitting the target, using $n = Q/e$) gave the kinetic energy per electron.
• These values agreed with $eV$, confirming the electrons behaved relativistically.