The Result
Bertozzi found that as the accelerating voltage got extremely high, the speed of the electrons stopped increasing and capped out just below 3.00×108 m s−13.00 \times 10^8 \text{ m s}^{-1}3.00×108 m s−1. Crucially, the calorimetry (heat measurement) at the target proved that the kinetic energy of the electrons was still increasing exactly as predicted by Ek=m0c21−v2c2−m0c2E_k = \frac{m_0 c^2}{\sqrt{1 - \frac{v^2}{c^2}}} - m_0 c^2Ek=1−c2v2m0c2−m0c2.
The energy hadn't disappeared; it had successfully gone into the electrons, increasing their relativistic mass and total energy, even though their speed had hardly increased.
Confusing the axes on the graphs
In the exam, pay close attention to whether the graph asks for Mass vs Speed or Kinetic Energy vs Speed. They look similar (both curve upwards to an asymptote at ccc), but Mass vs Speed starts at m0m_0m0 on the y-axis, whereas Kinetic Energy vs Speed starts at 000.
In the exam
- If a question asks for the kinetic energy of a particle moving faster than 0.1c0.1c0.1c, immediately write down Ek=E−E0E_k = E - E_0Ek=E−E0. Do not use 12mv2\frac{1}{2}mv^221mv2.
- To save time on your calculator when working with speeds given as fractions of ccc (e.g. 0.8c0.8c0.8c), you can calculate the Lorentz factor using just the fraction: 11−0.82\frac{1}{\sqrt{1 - 0.8^2}}1−0.821. The ccc cancels out inside the fraction v2c2\frac{v^2}{c^2}c2v2.
- If asked to describe Bertozzi's experiment, ensure you mention how both variables were measured: speed by time-of-flight between two detectors, and kinetic energy by calorimetry (measuring the heat generated when electrons strike a target).
Check yourself
- What does m0m_0m0 stand for, and when is a particle's mass equal to m0m_0m0?
- Why does the graph of relativistic mass against speed have a vertical asymptote at v=cv=cv=c?
- What two specific quantities did Bertozzi measure independently to prove the relativistic kinetic energy formula?
- An electron has a total energy of 3 MeV3\text{ MeV}3 MeV and a rest energy of 0.51 MeV0.51\text{ MeV}0.51 MeV. What is its kinetic energy?