Electron Speed Calculator
Calculate electron velocity from voltage, kinetic energy, or momentum using relativistic physics
Accelerating voltage applied to the electron
How to Use This Calculator
Select Calculation Mode
Choose whether you want to calculate from voltage (most common), kinetic energy in eV, or momentum.
Enter Your Value
Input the voltage in volts, kinetic energy in electron volts (eV), or momentum in kg·m/s depending on your selected mode.
Calculate
Click "Calculate Electron Speed" to get the velocity, Lorentz factor, and other relativistic parameters.
Interpret Results
Review the calculated velocity, Lorentz factor, kinetic energy, and momentum. For high speeds, relativistic effects become significant.
Formula
From Voltage: KE = eV
Total Energy: E = KE + mc²
Lorentz Factor: γ = E / (mc²)
Velocity: v = c√(1 - 1/γ²)
Momentum: p = γmv
where e is elementary charge, m is electron mass, c is speed of light
Example 1: From Voltage
Given: Voltage = 1000 V
KE = 1000 eV = 1.602 × 10⁻¹⁶ J
E_total = KE + mc² = 1.602 × 10⁻¹⁶ + 8.187 × 10⁻¹⁴ = 8.203 × 10⁻¹⁴ J
γ = 8.203 × 10⁻¹⁴ / 8.187 × 10⁻¹⁴ = 1.002
v = c√(1 - 1/1.002²) = 0.063c = 1.88 × 10⁷ m/s
Example 2: High Energy Electron
Given: Kinetic Energy = 1,000,000 eV (1 MeV)
KE = 1.602 × 10⁻¹³ J
γ = (KE + mc²) / mc² = (1.602 × 10⁻¹³ + 8.187 × 10⁻¹⁴) / 8.187 × 10⁻¹⁴ = 2.957
v = c√(1 - 1/2.957²) = 0.941c = 2.82 × 10⁸ m/s
About Electron Speed Calculator
The Electron Speed Calculator determines the velocity of an electron based on its voltage, kinetic energy, or momentum. Since electrons can reach speeds close to the speed of light, this calculator uses relativistic formulas to provide accurate results. It's essential for understanding electron behavior in particle accelerators, cathode ray tubes, and electron microscopes.
When to Use This Calculator
- Physics Education: Teaching students about electron acceleration and relativistic effects
- Electron Microscopy: Calculating electron velocities in electron microscopes
- Particle Physics: Determining electron speeds in accelerators and colliders
- Electronics: Understanding electron behavior in vacuum tubes and cathode ray tubes
- Research: Calculating electron parameters in experimental physics
Why Use Our Calculator?
- ✅ Relativistic Accuracy: Uses correct relativistic formulas for high-speed electrons
- ✅ Multiple Input Methods: Calculate from voltage, energy, or momentum
- ✅ Comprehensive Output: Provides velocity, Lorentz factor, momentum, and energy
- ✅ 100% Free: No registration or payment required
- ✅ Educational: Shows formulas and explains the physics
- ✅ Instant Results: Get accurate calculations immediately
Electron Acceleration
Electrons are accelerated by electric fields, gaining kinetic energy proportional to the voltage. At low energies (below about 50 keV), classical physics is sufficient. However, at higher energies, relativistic effects become important, and the electron's velocity approaches but never reaches the speed of light.
Common Applications
Electron Microscopes: Electron microscopes use accelerated electrons (typically 50-300 keV) to achieve high resolution. This calculator helps determine the electron velocity for a given accelerating voltage.
Particle Accelerators: In linear accelerators and synchrotrons, electrons are accelerated to very high energies (MeV to GeV range). This calculator helps understand the relativistic behavior of these particles.
Cathode Ray Tubes: Traditional CRT displays and oscilloscopes use electrons accelerated by voltages typically ranging from 1-30 kV. This calculator helps determine electron speeds in these devices.
Tips for Best Results
- For voltages below 50 kV, relativistic effects are small but still present
- At high energies (above 100 keV), relativistic effects become significant
- The Lorentz factor approaches 1 at low energies and increases rapidly at high energies
- Electron velocity never reaches the speed of light, even at infinite energy
- Use voltage mode for most practical applications involving electron acceleration
Frequently Asked Questions
Why do we need relativistic formulas for electrons?
Electrons have very small mass, so they can be accelerated to speeds approaching the speed of light with relatively modest energies. For example, a 1 MeV electron already travels at about 94% of the speed of light. Classical formulas would give incorrect results at these speeds.
What is the relationship between voltage and electron speed?
When an electron is accelerated through a voltage V, it gains kinetic energy KE = eV, where e is the elementary charge. The velocity depends on this kinetic energy and the electron's rest mass, using relativistic formulas.
Can an electron reach the speed of light?
No. According to special relativity, no object with mass can reach the speed of light. As an electron's energy increases, its velocity approaches but never reaches c. Most of the energy goes into increasing the electron's mass (via E = mc²) rather than its velocity.
What is the Lorentz factor?
The Lorentz factor (γ) quantifies how much time dilation and length contraction occur at a given velocity. It's calculated as γ = 1/√(1 - v²/c²) and approaches infinity as velocity approaches the speed of light.
When can I use classical formulas instead?
For electrons with kinetic energies below about 50 keV (velocities below about 0.4c), classical formulas give reasonable approximations. However, for accuracy, especially in research applications, relativistic formulas should always be used.
What is the typical speed of electrons in electron microscopes?
Electron microscopes typically use accelerating voltages of 50-300 kV, resulting in electron velocities of about 0.4c to 0.8c. At 100 kV, an electron travels at approximately 0.548c (54.8% of the speed of light).