Table of Contents
Fundamental Physical Constants and Their Significance
1. Introduction
Fundamental physical constants are quantities that appear in the mathematical laws of physics and are believed to have universal significance.
They describe fundamental properties of nature and establish relationships between physical phenomena.
Examples include:
- Speed of light in vacuum.
- Planck constant.
- Gravitational constant.
- Elementary electric charge.
- Boltzmann constant.
- Avogadro constant.
- Electron mass.
- Proton mass.
- Fine-structure constant.
Fundamental constants are important in physics, chemistry, astronomy, engineering, metrology and modern technology.
Some constants have exact numerical values by definition in the SI system, while others are experimentally measured and carry associated uncertainties.
2. Meaning of Physical Constants
A physical constant is a quantity that remains fixed within the relevant physical theory and system of units.
For example, Newton’s law of universal gravitation is expressed as:
[
F = G\frac{m_1m_2}{r^2}
]
Here, (G) is the universal gravitational constant.
Similarly, Einstein’s mass-energy relationship is:
[
E = mc^2
]
Here, (c) represents the speed of light in vacuum.
Constants allow physical laws to express quantitative relationships between different quantities.
3. Speed of Light in Vacuum
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The speed of light in vacuum is represented by (c).
Its exact SI value is:
[
\boxed{c = 299,792,458\ \text{m/s}}
]
It is approximately:
[
3.00 \times 10^8\ \text{m/s}
]
3.1 Significance
The speed of light is important in:
- Electromagnetic theory.
- Special and general relativity.
- Astronomy.
- Satellite communication.
- Optical technologies.
- Modern metrology.
3.2 Mass-Energy Equivalence
Einstein’s famous equation is:
[
E = mc^2
]
It establishes the equivalence of mass and energy.
A small amount of mass corresponds to a very large amount of energy because the speed of light is extremely large.
3.3 SI Definition of the Metre
The metre is defined by fixing the numerical value of the speed of light in vacuum.
Thus, the speed of light is an important foundation of modern length measurement.
4. Planck Constant
The Planck constant is represented by (h).
Its exact SI value is:
[
\boxed{h = 6.62607015 \times 10^{-34}\ \text{J s}}
]
4.1 Significance
The Planck constant is fundamental to quantum mechanics.
It relates the energy of a photon to its frequency:
[
E = hf
]
Where:
- (E) = photon energy.
- (h) = Planck constant.
- (f) = frequency.
Higher-frequency electromagnetic radiation carries higher-energy photons.
4.2 Reduced Planck Constant
The reduced Planck constant is represented by (\hbar).
[
\hbar = \frac{h}{2\pi}
]
It appears in many quantum-mechanical equations, including the Schrödinger equation.
4.3 Importance in Technology
The Planck constant is important in:
- Quantum physics.
- Semiconductor technology.
- Lasers.
- Atomic physics.
- Quantum computing research.
- Modern mass metrology.
The kilogram is defined through the fixed numerical value of the Planck constant.
5. Universal Gravitational Constant
The universal gravitational constant is represented by (G).
Its approximate CODATA 2022 value is:
[
\boxed{G = 6.67430 \times 10^{-11}\
\text{m}^3\text{kg}^{-1}\text{s}^{-2}}
]
Its measured uncertainty is approximately (1.5 \times 10^{-15}) in the same units.
5.1 Newton’s Law of Gravitation
[
F = G\frac{m_1m_2}{r^2}
]
Where:
- (F) = gravitational force.
- (G) = universal gravitational constant.
- (m_1) and (m_2) = masses.
- (r) = distance between their centres.
5.2 Significance
The gravitational constant is important in:
- Planetary motion.
- Satellite orbits.
- Astronomical calculations.
- Gravitational physics.
- Understanding the large-scale structure of the universe.
It should not be confused with (g), the local acceleration due to gravity near Earth’s surface.
The value of (g) is approximately:
[
g \approx 9.81\ \text{m/s}^2
]
Its actual value varies with location, altitude and local geological conditions.
6. Elementary Electric Charge
The elementary charge is represented by (e).
Its exact SI value is:
[
\boxed{e = 1.602176634 \times 10^{-19}\ \text{C}}
]
6.1 Significance
The elementary charge is the magnitude of the electric charge of:
- A proton, which has positive charge (+e).
- An electron, which has negative charge (-e).
It is a fundamental quantity in atomic and particle physics.
6.2 Coulomb’s Law
The magnitude of the electrostatic force between two point charges is:
[
F = \frac{1}{4\pi\varepsilon_0}
\frac{|q_1q_2|}{r^2}
]
Where:
- (F) = electrostatic force.
- (q_1) and (q_2) = electric charges.
- (r) = separation distance.
- (\varepsilon_0) = vacuum electric permittivity.
6.3 Applications
The elementary charge is important in:
- Electricity.
- Electronics.
- Electrochemistry.
- Semiconductor physics.
- Atomic structure.
- Particle physics.
7. Boltzmann Constant
The Boltzmann constant is represented by (k_B).
Its exact SI value is:
[
\boxed{k_B = 1.380649 \times 10^{-23}\ \text{J/K}}
]
7.1 Significance
The Boltzmann constant connects microscopic particle behaviour with macroscopic temperature.
It is fundamental to statistical mechanics and thermodynamics.
7.2 Thermal Energy
A characteristic thermal energy scale is:
[
E \sim k_BT
]
Where:
- (E) = characteristic thermal energy.
- (k_B) = Boltzmann constant.
- (T) = absolute temperature.
The expression (k_BT) is widely used in statistical physics.
7.3 Applications
The Boltzmann constant is important in:
- Thermodynamics.
- Statistical mechanics.
- Gas behaviour.
- Semiconductor physics.
- Thermal radiation.
- Molecular physics.
The kelvin is defined through the fixed numerical value of the Boltzmann constant.
8. Avogadro Constant
The Avogadro constant is represented by (N_A).
Its exact SI value is:
[
\boxed{N_A = 6.02214076 \times 10^{23}\ \text{mol}^{-1}}
]
8.1 Significance
The Avogadro constant specifies the number of elementary entities in one mole.
These entities may include:
- Atoms.
- Molecules.
- Ions.
- Electrons.
- Other specified particles.
8.2 Mole and Chemical Calculations
The relationship between the number of particles and amount of substance is:
[
N = nN_A
]
Where:
- (N) = number of specified entities.
- (n) = amount of substance in moles.
- (N_A) = Avogadro constant.
8.3 Importance
The Avogadro constant is essential in:
- Chemical calculations.
- Molecular chemistry.
- Stoichiometry.
- Atomic and molecular physics.
- Pharmaceutical research.
- Materials science.
The mole is defined by fixing the numerical value of the Avogadro constant.
9. Fine-Structure Constant
The fine-structure constant is represented by (\alpha).
It is a dimensionless physical constant.
Its approximate value is:
[
\boxed{\alpha \approx \frac{1}{137.036}}
]
9.1 Significance
The fine-structure constant characterizes the strength of the electromagnetic interaction.
It is important in:
- Atomic physics.
- Quantum electrodynamics.
- Spectroscopy.
- Quantum theory.
- Studies of fundamental interactions.
Because it is dimensionless, its numerical value does not depend on the choice of measurement units.
10. Electron Mass
The electron mass is represented by (m_e).
Its approximate CODATA 2022 value is:
[
\boxed{m_e \approx 9.1093837 \times 10^{-31}\ \text{kg}}
]
10.1 Significance
The electron is a fundamental subatomic particle with a negative electric charge.
Its mass is important in:
- Atomic structure.
- Chemical bonding.
- Electricity.
- Semiconductor physics.
- Quantum mechanics.
- Electron microscopy.
11. Proton Mass
The proton mass is represented by (m_p).
Its approximate CODATA 2022 value is:
[
\boxed{m_p \approx 1.6726219 \times 10^{-27}\ \text{kg}}
]
11.1 Significance
A proton is a positively charged particle found in the nucleus of an atom.
The proton mass is important in:
- Nuclear physics.
- Atomic structure.
- Particle physics.
- Chemistry.
- Nuclear reactions.
The proton is not a fundamental elementary particle; it is composed of quarks held together by the strong interaction.
12. Neutron Mass
The neutron mass is represented by (m_n).
Its approximate value is:
[
\boxed{m_n \approx 1.6749275 \times 10^{-27}\ \text{kg}}
]
12.1 Significance
A neutron is a subatomic particle with no net electric charge.
It is found in the nuclei of most atoms, with ordinary hydrogen-1 being the major exception.
Neutrons play important roles in:
- Nuclear stability.
- Nuclear reactions.
- Nuclear fission.
- Nuclear fusion.
- Isotope formation.
13. Vacuum Permittivity
Vacuum permittivity is represented by (\varepsilon_0).
Its approximate CODATA 2022 value is:
[
\boxed{\varepsilon_0 \approx
8.8541878 \times 10^{-12}\ \text{F/m}}
]
13.1 Significance
Vacuum permittivity describes an important electrical property of vacuum in electromagnetic theory.
It appears in:
- Coulomb’s law.
- Maxwell’s equations.
- Electromagnetic wave theory.
- Capacitance calculations.
Unlike its former SI status, the vacuum permittivity is no longer an exact defined constant following the 2019 SI revision. Its value is experimentally determined in the current SI.
14. Vacuum Permeability
Vacuum permeability is represented by (\mu_0).
Its approximate value is:
[
\boxed{\mu_0 \approx
1.2566371 \times 10^{-6}\ \text{N/A}^2}
]
14.1 Significance
Vacuum permeability describes an important magnetic property of vacuum in classical electromagnetism.
It is used in:
- Magnetic field calculations.
- Electromagnetic theory.
- Inductance calculations.
- Electrical engineering.
In the revised SI, vacuum permeability is no longer an exact defined constant and is related to experimentally determined electromagnetic quantities.
15. Gas Constant
The molar gas constant is represented by (R).
Its exact SI value is:
[
\boxed{R = 8.314462618\ldots\ \text{J mol}^{-1}\text{K}^{-1}}
]
15.1 Ideal Gas Equation
The ideal gas equation is:
[
PV=nRT
]
Where:
- (P) = pressure.
- (V) = volume.
- (n) = amount of gas in moles.
- (R) = molar gas constant.
- (T) = absolute temperature.
15.2 Relationship with Boltzmann Constant
The molar gas constant is related to the Boltzmann constant through the Avogadro constant:
[
R=N_Ak_B
]
15.3 Applications
The gas constant is used in:
- Thermodynamics.
- Chemistry.
- Gas calculations.
- Chemical engineering.
- Atmospheric science.
16. Stefan-Boltzmann Constant
The Stefan-Boltzmann constant is represented by (\sigma).
Its approximate value is:
[
\boxed{\sigma \approx
5.6703744 \times 10^{-8}\
\text{W m}^{-2}\text{K}^{-4}}
]
16.1 Stefan-Boltzmann Law
The total radiant energy emitted per unit surface area per unit time by an ideal blackbody is proportional to the fourth power of its absolute temperature.
[
j^* = \sigma T^4
]
Where:
- (j^*) = emitted radiant power per unit area.
- (\sigma) = Stefan-Boltzmann constant.
- (T) = absolute temperature.
For a real surface, an emissivity factor is included under the appropriate conditions.
16.2 Applications
The constant is important in:
- Thermal radiation.
- Astrophysics.
- Solar physics.
- Climate science.
- Heat-transfer engineering.
17. Wien’s Displacement Constant
Wien’s displacement constant is represented by (b).
Its approximate value is:
[
\boxed{b \approx 2.89777196 \times 10^{-3}\ \text{m K}}
]
17.1 Wien’s Displacement Law
The wavelength at which a blackbody spectrum has its maximum intensity is inversely proportional to its absolute temperature.
[
\lambda_{\max}T=b
]
Where:
- (\lambda_{\max}) = wavelength of maximum spectral intensity.
- (T) = absolute temperature.
- (b) = Wien’s displacement constant.
17.2 Applications
Wien’s law is useful in:
- Determining the surface temperature of stars.
- Studying thermal radiation.
- Infrared astronomy.
- Understanding the relationship between temperature and emitted radiation.
18. Summary of Important Physical Constants
The following table gives important constants and their approximate or exact values.
| Constant | Symbol | Value |
|---|---|---|
| Speed of light in vacuum | (c) | (299,792,458\ \text{m/s}) exactly |
| Planck constant | (h) | (6.62607015 \times 10^{-34}\ \text{J s}) exactly |
| Gravitational constant | (G) | (6.67430 \times 10^{-11}\ \text{m}^3\text{kg}^{-1}\text{s}^{-2}) approximately |
| Elementary charge | (e) | (1.602176634 \times 10^{-19}\ \text{C}) exactly |
| Boltzmann constant | (k_B) | (1.380649 \times 10^{-23}\ \text{J/K}) exactly |
| Avogadro constant | (N_A) | (6.02214076 \times 10^{23}\ \text{mol}^{-1}) exactly |
| Fine-structure constant | (\alpha) | Approximately (1/137.036) |
| Electron mass | (m_e) | Approximately (9.1093837 \times 10^{-31}\ \text{kg}) |
| Proton mass | (m_p) | Approximately (1.6726219 \times 10^{-27}\ \text{kg}) |
| Neutron mass | (m_n) | Approximately (1.6749275 \times 10^{-27}\ \text{kg}) |
| Vacuum permittivity | (\varepsilon_0) | Approximately (8.8541878 \times 10^{-12}\ \text{F/m}) |
| Vacuum permeability | (\mu_0) | Approximately (1.2566371 \times 10^{-6}\ \text{N/A}^2) |
| Molar gas constant | (R) | (8.314462618\ldots\ \text{J mol}^{-1}\text{K}^{-1}) |
| Stefan-Boltzmann constant | (\sigma) | Approximately (5.6703744 \times 10^{-8}\ \text{W m}^{-2}\text{K}^{-4}) |
| Wien’s displacement constant | (b) | Approximately (2.89777196 \times 10^{-3}\ \text{m K}) |
Note: Approximate measured values are rounded for educational use. The exact values of SI-defining constants are exact by definition. Numerical values of measured constants can be updated as experimental knowledge improves.
19. The 2019 Revision of the SI System
A major change in the International System of Units came into effect on 20 May 2019.
The revised SI system defines all seven base units through fixed numerical values of seven fundamental constants.
These constants are:
- Caesium-133 hyperfine transition frequency.
- Speed of light in vacuum.
- Planck constant.
- Elementary charge.
- Boltzmann constant.
- Avogadro constant.
- Luminous efficacy of monochromatic radiation of specified frequency.
19.1 Importance of the Revision
The revision:
- Reduced dependence on physical artefacts.
- Established a more fundamental basis for measurement.
- Improved long-term stability.
- Strengthened international measurement consistency.
- Connected SI units with fundamental properties of nature.
19.2 Fundamental Constants and SI Base Units
| SI base unit | Defining constant |
|---|---|
| Second | Caesium-133 transition frequency |
| Metre | Speed of light |
| Kilogram | Planck constant |
| Ampere | Elementary charge |
| Kelvin | Boltzmann constant |
| Mole | Avogadro constant |
| Candela | Luminous efficacy of specified monochromatic radiation |
20. Importance of Fundamental Physical Constants
Fundamental constants are essential because they:
- Establish quantitative relationships in physical laws.
- Support the development of theoretical physics.
- Help define SI measurement units.
- Enable precise scientific measurements.
- Support atomic and molecular calculations.
- Improve astronomical models.
- Help test the validity of physical theories.
- Support advanced technologies.
- Provide reference values for scientific research.
- Help scientists investigate the fundamental structure of nature.
21. Conclusion
Fundamental physical constants are essential numerical quantities that underpin the laws of nature and modern measurement systems.
Constants such as the speed of light, Planck constant, gravitational constant, elementary charge, Boltzmann constant and Avogadro constant are central to physics, chemistry, engineering and technology.
Their importance extends beyond theoretical science because they also form the foundation of accurate measurement, modern instrumentation and advanced technological applications.
36 Key Takeaways for Exams
- A physical constant is a quantity with a fixed value within a specified physical theory and system of units.
- The speed of light in vacuum is represented by (c).
- The exact speed of light in vacuum is 299,792,458 metres per second.
- Einstein’s mass-energy equation is (E=mc^2).
- The speed of light defines the SI metre.
- The Planck constant is represented by (h).
- The exact Planck constant is (6.62607015 \times 10^{-34}\ \text{J s}).
- Photon energy is expressed as (E=hf).
- The reduced Planck constant is represented by (\hbar).
- The universal gravitational constant is represented by (G).
- The approximate value of (G) is (6.67430 \times 10^{-11}\ \text{m}^3\text{kg}^{-1}\text{s}^{-2}).
- Newton’s law of gravitation contains the universal gravitational constant.
- The local acceleration due to gravity is represented by (g), not (G).
- The elementary charge is represented by (e).
- The exact elementary charge is (1.602176634 \times 10^{-19}) coulomb.
- A proton has positive charge, while an electron has negative charge.
- The Boltzmann constant is represented by (k_B).
- The exact Boltzmann constant is (1.380649 \times 10^{-23}\ \text{J/K}).
- The Boltzmann constant connects microscopic physics with thermodynamics.
- The Avogadro constant is represented by (N_A).
- One mole contains exactly (6.02214076 \times 10^{23}) specified elementary entities.
- The Avogadro constant is essential for chemical calculations.
- The fine-structure constant is a dimensionless physical constant.
- The fine-structure constant is approximately equal to 1/137.036.
- The electron is a negatively charged subatomic particle.
- The proton is a positively charged particle found in atomic nuclei.
- The neutron has no net electric charge.
- Vacuum permittivity is represented by (\varepsilon_0).
- Vacuum permeability is represented by (\mu_0).
- The molar gas constant is represented by (R).
- The ideal gas equation is (PV=nRT).
- The molar gas constant is related to the Boltzmann and Avogadro constants through (R=N_Ak_B).
- The Stefan-Boltzmann law relates blackbody radiation to the fourth power of absolute temperature.
- Wien’s displacement law relates peak wavelength inversely to absolute temperature.
- The revised SI system came into effect on 20 May 2019.
- The revised SI defines all seven base units through fixed numerical values of seven defining constants.