Tru Physics

Heaviside-Lorentz Units

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Introduction

Heaviside-Lorentz units (named for O. Heaviside and H.A. Lorentz) constitute a particular extension of CGS units which are most often useful in theoretical electrodynamics. As a practical system of units, Heaviside-Lorentz falls short due to most measureable quantities being excessively small or big for intuition. Neverless, Heaviside-Lorentz showcases the simplicity and beauty of the electromagnetic equations more than both SI and CGS systems.

The primary motivation behind Heaviside-Lorentz units is the setting of the vacuum permittivity (\varepsilon_0) and vacuum permeability (\mu_0) both to 1 while retaining factors of 4 \pi found in the SI equations. Factors of c (the speed of light) are included as needed to fix units (just as with CGS units).

\varepsilon_0=1

\mu_0=1

Units of Primary Electromagnetic Quantities

Fields

The units of \vec{E}, \vec{B}, \vec{D}, \vec{H}, \vec{P}, and \vec{M} are the same in Heaviside-Lorentz. Units for these quantities are:

\sqrt{\dfrac{\text{g}}{\text{cm}}}\cdot\dfrac{1}{\text{s}}

  • In order to obtain units of \vec{A} and \phi: multiply by units of length (\text{cm}).
  • In order to obtain units of q, \Phi_E, and \Phi_B: multiply by units of length squared (\text{cm}^2).

Charge and Flux

The units of q, \Phi_E, and \Phi_B are also the same in Heaviside-Lorentz. Units for these quantities are:

\sqrt{\text{g}\cdot\text{cm}}\cdot\dfrac{\text{cm}}{\text{s}} \text{ or } \sqrt{\text{g}\cdot\text{cm}^3}\cdot\dfrac{1}{\text{s}}

  • In order to obtain units of \vec{A} and \phi: divide by units of length (\text{cm}).
  • In order to obtain units of \vec{E}, \vec{B}, \vec{D}, \vec{H}, \vec{P}, and \vec{M}: divide by units of length squared (\text{cm}^2).

Potentials

In like fashion, the units of \vec{A} and \phi are the same in Heaviside-Lorentz. Units for these quantities are:

\sqrt{\text{g}\cdot\text{cm}}\cdot\dfrac{1}{\text{s}}

  • In order to obtain units of q, \Phi_E, and \Phi_B: multiply by units of length (\text{cm}).
  • In order to obtain units of \vec{E}, \vec{B}, \vec{D}, \vec{H}, \vec{P}, and \vec{M}: divide by units of length (\text{cm}).

The Maxwell Equations

The Maxwell equations (microscopic form) can be written elegantly in Heaviside-Lorentz units.

\vec{\nabla}\cdot\vec{E}(\vec{r},t)=\rho(\vec{r},t)

\vec{\nabla}\cdot\vec{B}(\vec{r},t)=0

\vec{\nabla}\times\vec{E}(\vec{r},t)=-\dfrac{1}{c}\dfrac{\partial}{\partial t}\vec{B}(\vec{r},t)

\vec{\nabla}\times\vec{B}(\vec{r},t)=\dfrac{1}{c}\vec{J}(\vec{r},t)+\dfrac{1}{c}\dfrac{\partial}{\partial t}\vec{E}(\vec{r},t)

The Lorentz Force Law

There is a minor modification to the Lorentz force law compared to the well-known SI version. We must include a factor of 1/c in the magnetic force term in order to obtain proper Heaviside-Lorentz units. For a particle with trajectory \vec{r}_p(t), we write:

\vec{F}(t)=q[\vec{E}(\vec{r}_p(t),t)+\dfrac{1}{c}\dfrac{d \vec{r}_p(t)}{dt}\times\vec{B}(\vec{r}_p(t),t)]

Other Electromagnetic Quantities

Force:

[\vec{F}]=\dfrac{\text{g}\cdot\text{cm}}{\text{s}^2}

Energy:

[E]=\dfrac{\text{g}\cdot\text{cm}^2}{\text{s}^2}

Power:

[P]=\dfrac{\text{g}\cdot\text{cm}^2}{\text{s}^3}

Electric Diploe Moment:

[\vec{d}]=\sqrt{\text{g}\cdot\text{cm}}\cdot\dfrac{\text{cm}^2}{\text{s}}

Magnetic Diploe Moment:

[\vec{m}]=\sqrt{\text{g}\cdot\text{cm}}\cdot\dfrac{\text{cm}^2}{\text{s}}

Electric Susceptibility:

[\chi_e]=1 (unitless)

Magnetic Susceptibility:

[\chi_m]=1 (unitless)

Resistivity:

[\rho]=s

Conductivity:

[\sigma]=\dfrac{1}{s}

Voltage:

[V]=\dfrac{\sqrt{\text{g}\cdot\text{cm}}}{\text{s}}

Current:

[I]=\sqrt{\text{g}\cdot\text{cm}}\cdot\dfrac{\text{cm}}{\text{s}^2}

Resistance:

[R]=\dfrac{\text{s}}{\text{cm}}

Capacitance:

[C]=\text{cm}

Inductance:

[L]=\dfrac{\text{s}^2}{\text{cm}}

Bulk Current Density:

[\vec{J}]=\sqrt{\dfrac{\text{g}}{\text{cm}}}\cdot\dfrac{1}{\text{s}^2}

Volume Charge Density:

[\rho]=\sqrt{\dfrac{\text{g}}{\text{cm}}}\cdot\dfrac{1}{\text{cm}\cdot\text{s}}

Surface Charge Density:

[\rho]=\sqrt{\dfrac{\text{g}}{\text{cm}}}\cdot\dfrac{1}{\text{s}}

Line Charge Density:

[\lambda]=\sqrt{\text{g}\cdot\text{cm}}\cdot\dfrac{1}{\text{s}}

This is not intended to be a full exploration of every electromagnetic quantity in Heaviside-Lorentz units. Rather, this is a starting point for further study and use of this unit system. NASA has produced a useful resource for help determining the units of other quantities of interest.

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