Speed of electricity

The speed of electromagnetic waves in a low-loss dielectric is given by

${\displaystyle \quad v={\frac {1}{\sqrt {\epsilon \mu }}}={\frac {c}{\sqrt {\epsilon _{r}\mu _{r}}}}}$ .[1]^:346

where

${\displaystyle \quad c}$ = speed of light in vacuum.

${\displaystyle \quad \mu _{0}}$ = the permeability of free space = 4π x 10⁻⁷ H/m.

${\displaystyle \quad \mu _{r}}$ = relative magnetic permeability of the material. Usually in good dielectrics, eg. vacuum, air, Teflon, ${\displaystyle \mu _{r}=1}$.

${\displaystyle \quad \mu }$ = ${\displaystyle \mu _{r}}$ ${\displaystyle \mu _{0}}$.

${\displaystyle \quad \epsilon _{0}}$ = the permitivity of free space = 8.854 x 10⁻¹² F/m.

${\displaystyle \quad \epsilon _{r}}$ = relative permitivity of the material. Usually in good conductors eg. copper, silver, gold, ${\displaystyle \epsilon _{r}=1}$.

${\displaystyle \quad \epsilon }$ = ${\displaystyle \epsilon _{r}}$ ${\displaystyle \epsilon _{0}}$.

The speed of electromagnetic waves in a good conductor is given by

${\displaystyle \quad v={\sqrt {\frac {2\omega }{\sigma \mu }}}={\sqrt {\frac {4\pi }{\sigma _{c}\mu _{0}}}}{\sqrt {\frac {f}{\sigma _{r}\mu _{r}}}}\approx \left(0.41~\mathrm {m/s} \right){\sqrt {\frac {f}{\sigma _{r}\mu _{r}}}}}$.[1]^:360 [2]^:142 [3]^:50–52

where

${\displaystyle \quad f}$ = frequency.

${\displaystyle \quad \omega }$ = angular frequency = 2πf.

${\displaystyle \quad \sigma _{c}}$ = conductivity of annealed copper = 5.96×10⁷ S/m.

${\displaystyle \quad \sigma _{r}}$ = conductivity of the material relative to the conductivity of copper. For hard drawn copper ${\displaystyle \sigma _{r}}$ may be as low as 0.97.

${\displaystyle \quad \sigma }$ = ${\displaystyle \sigma _{r}\sigma _{c}}$.

and permeability is defined as above in § Speed of electromagnetic waves in good dielectrics

${\displaystyle \quad \mu _{0}}$ = the permeability of free space = 4π x 10⁻⁷ H/m.

${\displaystyle \quad \mu _{r}}$ = relative magnetic permeability of the material. Magnetically conductive materials such as copper typically have a ${\displaystyle \mu _{r}}$ near 1.

${\displaystyle \quad \mu }$ = ${\displaystyle \mu _{r}}$ ${\displaystyle \mu _{0}}$.

In copper at 60 Hz, ${\displaystyle v\approx }$ 3.2 m/s. As a consequence of Snell's Law and the extremely low speed, electromagnetic waves always enter good conductors in a direction that is within a milliradian of normal to the surface, regardless of the angle of incidence. This velocity is the speed with which electromagnetic waves penetrate into the conductor and is not the drift velocity of the conduction electrons.

In the theoretical investigation of electric circuits, the velocity of propagation of the electromagnetic field through space is usually not considered; the field is assumed, as a precondition, to be present throughout space. The magnetic component of the field is considered to be in phase with the current, and the electric component is considered to be in phase with the voltage. The electric field starts at the conductor, and propagates through space at the velocity of light (which depends on the material it is traveling through). Note that the electromagnetic fields do not move through space. It is the electromagnetic energy that moves, the corresponding fields simply grow and decline in a region of space in response to the flow of energy. At any point in space, the electric field corresponds not to the condition of the electric energy flow at that moment, but to that of the flow at a moment earlier. The latency is determined by the time required for the field to propagate from the conductor to the point under consideration. In other words, the greater the distance from the conductor, the more the electric field lags.[4]

Since the velocity of propagation is very high – about 300,000 kilometers per second – the wave of an alternating or oscillating current, even of high frequency, is of considerable length. At 60 cycles per second, the wavelength is 5,000 kilometers, and even at 100,000 hertz, the wavelength is 3 kilometers. This is a very large distance compared to those typically used in field measurement and application.[4]

The important part of the electric field of a conductor extends to the return conductor, which usually is only a few feet distant. At greater distance, the aggregate field can be approximated by the differential field between conductor and return conductor, which tend to cancel. Hence, the intensity of the electric field is usually inappreciable at a distance which is still small compared to the wavelength. Within the range in which an appreciable field exists, this field is practically in phase with the flow of energy in the conductor. That is, the velocity of propagation has no appreciable effect unless the return conductor is very distant, or entirely absent, or the frequency is so high that the distance to the return conductor is an appreciable portion of the wavelength.[4]