Heaviside–Lorentz units
Rationalized CGS-based units eliminating 4π from Maxwell's equations.
Heaviside–Lorentz units (also called Lorentz–Heaviside units) are a system of units and quantities that extend the CGS system with a particular set of equations defining electromagnetic quantities. Named for Oliver Heaviside and Hendrik Antoon Lorentz, they normalize the electric constant ε₀ and magnetic constant μ₀ to 1 and revise Maxwell's equations to use the speed of light c, removing explicit factors of 4π. This rationalized system is often used in relativistic calculations and particle physics, and is particularly convenient in quantum field theory and string theory.
The system arose from a suggestion by Heaviside in 1882 to remove the irrational appearance of 4π from formulas lacking circular or spherical symmetry, such as the capacitance of a parallel plate capacitor. Heaviside elaborated this normalization in his 1893 book *Electromagnetic Theory*. Lorentz later popularized the system, using it in his 1906 lectures published as *The Theory of Electrons*, where he credited Heaviside. Like the CGS-Gaussian system, Heaviside–Lorentz units use length, mass, and time as base dimensions, so all electromagnetic units derive from these. However, unlike the Gaussian system, it is rationalized, meaning no factors of 4π appear explicitly in Maxwell’s equations. This rationalization makes it appealing in quantum field theory, as the Lagrangian lacks such factors. In the Heaviside–Lorentz system, Coulomb’s equation defines charge as F = q₁q₂/(4πr²), whereas in Gaussian units it is F = q₁q₂/r². Consequently, the Heaviside–Lorentz unit of charge is √(4π) times larger than the Gaussian unit. Conversion to SI units is possible using the constants ε₀ and μ₀; for example, SI charge q_SI = q_HL/√(4πε₀). The system’s removal of 4π from Gauss’s law clarifies that the inverse-square force law arises from field spreading over a sphere’s surface, facilitating extension to other dimensions, such as in string theory with more than three spatial dimensions.
- field
- Electromagnetism, unit systems
- known_for
- Rationalized electromagnetic units without 4π in Maxwell's equations
- associated_with
- Oliver Heaviside, Hendrik Antoon Lorentz
Lore & Background
Heaviside–Lorentz units, also known as Lorentz–Heaviside units, are a system of units and quantities that extend the CGS system with a specific set of equations defining electromagnetic quantities. They are named for Oliver Heaviside and Hendrik Antoon Lorentz. Like the CGS-Gaussian system, the electric constant and magnetic constant do not appear explicitly in the defining equations, as they are incorporated into the electromagnetic quantities themselves. The system can be understood as normalizing these constants while revising Maxwell's equations to use the speed of light. A key defining characteristic is that the system is rationalized, meaning no factors of 4π appear explicitly in Maxwell's equations. This rationalization makes the system appealing in quantum field theory, as the Lagrangian lacks such factors. Consequently, electromagnetic quantities in this system differ from those in Gaussian units by factors of √4π in the definitions of electric and magnetic fields and electric charge. Heaviside–Lorentz units are often used in relativistic calculations and in particle physics, and are particularly convenient for calculations in spatial dimensions greater than three, such as in string theory. The system uses length, mass, and time as base dimensions, so all electric and magnetic units are expressible in these terms. The unit of charge in the Heaviside–Lorentz system is √4π times larger than the corresponding Gaussian quantity.
Reader's Guide
Heaviside–Lorentz units are significant because they rationalize electromagnetic equations, removing factors of 4π that appear in CGS-Gaussian and other systems. This rationalization partly explains their appeal in quantum field theory, where the Lagrangian underlying the theory does not have any factors of 4π when this system is used. The system is often used in relativistic calculations and particle physics, and is particularly convenient when performing calculations in spatial dimensions greater than three, such as in string theory. In the Heaviside–Lorentz system, electromagnetic quantities differ by factors of √4π in the definitions of electric and magnetic fields and of electric charge compared to Gaussian units. For example, the HL unit of charge is √4π times larger than the corresponding Gaussian quantity. The system uses length–mass–time dimensions, with Coulomb's equation given as F = q₁q₂/(4πr²). Conversion to SI units involves the vacuum permittivity ε₀ and permeability μ₀, with SI charge expressed as √(ε₀L³M/T²). The system's legacy lies in providing a cleaner mathematical framework for theoretical physics, particularly in contexts where spherical symmetry is not inherent.
Did You Know?
- Heaviside–Lorentz units normalize ε₀ = 1 and μ₀ = 1 while revising Maxwell's equations to use the speed of light c.
- The system is rationalized, so no factors of 4π appear explicitly in Maxwell's equations.
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