Electron Speed Calculator
Classical vs. Relativistic Speeds from Kinetic Energy
Introduction
This calculator determines the speed of an electron given its kinetic energy (KE). It provides results from two perspectives:
- Classical Mechanics: Using the formula KE = 0.5mv2. This is accurate for speeds much lower than the speed of light.
- Special Relativity: Using KE = (γ - 1)m0c2. This is accurate at all speeds, including those approaching the speed of light.
By comparing these results, you can see when classical physics breaks down and relativistic effects become dominant.
How to use:
- Enter the Kinetic Energy of the electron.
- Select the appropriate unit for the kinetic energy (eV, keV, MeV, or Joules).
- Click "Calculate Speed".
Theory & Formulas
Constants Used:
| Constant | Symbol | Value |
|---|---|---|
| Electron Rest Mass | m0 | 9.1093837 × 10-31 kg |
| Speed of Light | c | 299,792,458 m/s |
| Elementary Charge | e | 1.602176634 × 10-19 C |
1. Classical Kinetic Energy (KEclass):
In Newtonian physics, the kinetic energy of an object is:
From this, the classical speed (vclass) can be derived:
This formula becomes inaccurate as v approaches c.
2. Relativistic Concepts:
Rest Energy (E0): An electron, even at rest, has energy due to its mass:
Lorentz Factor (γ - Gamma): A key factor in special relativity:
Where v is the relativistic speed.
Total Relativistic Energy (E):
Relativistic Kinetic Energy (KErel): The energy of motion in relativity is the total energy minus the rest energy:
3. Deriving Relativistic Speed (vrel):
From the KErel formula, we can find γ:
Then, by rearranging the Lorentz factor formula, we find vrel: