Electron Speed Calculator
Calculate classical and relativistic electron velocities in electric fields with precise physics formulas
Calculate Electron Speed
Electric potential difference accelerating the electron (1.10e+4 V)
Preset Scenarios
Electron Speed Results
Relativistic Parameters
Energy Analysis
Formulas Used
⚠️ Significant Relativistic Effects
The electron is moving at 20.4% of the speed of light. Relativistic effects are significant, and the classical formula gives a1.6% error.
Velocity Analysis
Example Calculation
CRT Monitor Example
Accelerating potential: 11 kV
Application: Cathode ray tube (CRT) monitor
Physical significance: Electrons accelerated from cathode to phosphor screen
Calculation Results
Classical velocity: vn = √(2 × 1.602×10⁻¹⁹ × 11,000 / 9.109×10⁻³¹) = 62,205 km/s
Relativistic velocity: vrel = 61,221 km/s
Difference: 984 km/s (1.6% error in classical calculation)
Conclusion: Small but measurable relativistic correction needed
Electron Properties
Key Physics Concepts
Work-Energy Theorem
Work done by electric field equals change in kinetic energy
Classical Mechanics
Valid for velocities much less than speed of light
Special Relativity
Accounts for mass-energy equivalence at high speeds
Electric Acceleration
Charged particles gain kinetic energy in electric fields
Real-World Applications
CRT Displays
Electron beam acceleration for screen phosphors
Electron Microscopy
High-resolution imaging with electron beams
Medical X-rays
Electron acceleration for X-ray production
Particle Physics
High-energy electron beam experiments
Electronic Devices
Vacuum tubes and electron guns
Understanding Electron Speed in Electric Fields
Classical vs. Relativistic Physics
When electrons are accelerated by electric fields, they gain kinetic energy. For low voltages, classical mechanics provides accurate results. However, as the accelerating potential increases, electrons reach velocities where relativistic effects become significant.
Physical Derivation
The work done by an electric field on an electron is W = eVa, where e is the elementary charge and Va is the accelerating potential. By conservation of energy, this work equals the electron's kinetic energy gain.
When to Use Each Formula
Classical Formula: v = √(2eV/m)
Use when: v < 0.1c (velocity less than 30,000 km/s)
Relativistic Formula: v = c√(1 - 1/(1 + eV/(mc²))²)
Use when: v ≥ 0.1c (high-energy applications)
Historical Context
The discovery of electrons by J.J. Thomson in 1897 used cathode ray tubes with electric and magnetic fields. Modern applications range from simple electronics to cutting-edge particle accelerators where electrons reach 99.9% of the speed of light.
Velocity Regimes and Applications
Low Energy
v < 0.01c
Electronics, simple vacuum tubes. Classical mechanics very accurate.
Moderate Energy
0.01c ≤ v < 0.1c
CRT displays, electron microscopes. Small relativistic corrections.
High Energy
0.1c ≤ v < 0.5c
Medical accelerators, research equipment. Relativistic effects important.
Ultra-High Energy
v ≥ 0.5c
Particle physics experiments. Fully relativistic treatment required.