Virial Theorem Calculator
Virial Theorem Calculator computes kinetic and potential energies for a particle in a potential, verifying the virial theorem.
Formulas Used in Virial Theorem Calculator
The calculator applies the virial theorem for a single particle:
Virial Theorem:
\[ \langle T \rangle = -\frac{1}{2} \sum_i \langle \mathbf{r}_i \cdot \mathbf{F}_i \rangle \]Harmonic Potential:
\[ V = \frac{1}{2} k r^2, \quad \mathbf{F} = -k \mathbf{r}, \quad \langle T \rangle = \langle V \rangle = \frac{3}{2} k_B T \]Gravitational Potential:
\[ V = -\frac{G M m}{r}, \quad \mathbf{F} = -\frac{G M m}{r^2} \hat{r}, \quad \langle T \rangle = -\frac{1}{2} \langle V \rangle \]Where:
- \(\langle T \rangle\): Average kinetic energy (J)
- \(\langle V \rangle\): Average potential energy (J)
- \(k_B\): Boltzmann constant (\(1.380649 \times 10^{-23} \, \text{J/K}\))
- \(T\): Temperature (K)
- \(k\): Spring constant (N/m)
- \(G\): Gravitational constant (\(6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}\))
- \(M\): Central mass (kg)
- \(m\): Particle mass (kg)
- \(r\): Distance (m)
Example Calculation
Example: Harmonic Potential, \(k = 1 \, \text{N/m}, T = 300 \, \text{K}\)
\[
\langle T \rangle = \frac{3}{2} k_B T = \frac{3}{2} \times 1.380649 \times 10^{-23} \times 300 \approx 6.212921 \times 10^{-21} \, \text{J}
\]
\[
\langle V \rangle = \langle T \rangle \approx 6.212921 \times 10^{-21} \, \text{J}
\]
Result: \(\langle T \rangle \approx 6.212921 \times 10^{-21} \, \text{J}, \langle V \rangle \approx 6.212921 \times 10^{-21} \, \text{J}\).