Orders of Magnitude in Electromagnetics

Why this page mattersMany mistakes are detectable before calculation. If you know roughly what a wavelength, noise floor, magnetic field, skin depth or RF period should be, a unit error becomes obvious immediately.

Frequency, period and wavelength

In free space,

\[\lambda=\frac{c}{f},\qquad T=\frac1f.\]

Frequency	Period	Free-space wavelength	Mental picture
1 Hz	1 s	$3\times10^8$ m	planetary scale
1 kHz	1 ms	300 km	low-frequency fields
1 MHz	1 μs	300 m	AM/HF/VHF transition scale
10 MHz	100 ns	30 m	HF
100 MHz	10 ns	3 m	VHF
1 GHz	1 ns	30 cm	microwave / wireless
10 GHz	100 ps	3 cm	radar / microwave
100 GHz	10 ps	3 mm	mmWave
1 THz	1 ps	300 μm	THz
100 THz	10 fs	3 μm	infrared
500 THz	2 fs	600 nm	visible-light scale

A very useful RF rule is

\[\boxed{\lambda(\mathrm{cm})\approx\frac{30}{f(\mathrm{GHz})}}.\]

Electrical size

Physical size by itself is not enough. What matters is

\[\frac{L}{\lambda}\quad\text{or}\quad ka=\frac{2\pi a}{\lambda}.\]

$L\ll\lambda$: lumped/quasistatic intuition often works.
$L\sim\lambda$: phase variation, standing waves and radiation matter.
$L\gg\lambda$: the object is electrically large; asymptotic or large full-wave methods may be needed.

ExampleA 10-cm cable is electrically tiny at 1 MHz ($\lambda\approx300$ m), but one third of a wavelength at 1 GHz ($\lambda\approx30$ cm). The same physical wire changes from a lumped interconnect into a distributed electromagnetic structure.

RF power and dBm intuition

Power	dBm
1 W	+30 dBm
100 mW	+20 dBm
10 mW	+10 dBm
1 mW	0 dBm
100 μW	−10 dBm
1 μW	−30 dBm
1 nW	−60 dBm
1 pW	−90 dBm
1 fW	−120 dBm

Remember:

\[+3\ \mathrm{dB}\approx2\times\text{ power},\qquad +10\ \mathrm{dB}=10\times\text{ power}.\]

A negative dBm value means less than 1 mW, not negative physical power.

Thermal-noise scale

Near room temperature,

\[N_0\approx-174\ \mathrm{dBm/Hz}.\]

Ignoring noise figure for the moment:

Bandwidth	Thermal noise near 290 K
1 Hz	−174 dBm
1 kHz	−144 dBm
100 kHz	−124 dBm
1 MHz	−114 dBm
10 MHz	−104 dBm
100 MHz	−94 dBm

Each factor of 10 in bandwidth raises integrated noise by 10 dB.

Engineering realityThe actual receiver floor also includes noise figure, loss before the LNA, gain distribution, phase noise, spurs, ADC quantization and environmental interference.

Magnetic-field scales

Field	Approximate scale	Context
femtotesla	$10^{-15}$ T	ultimate atomic/biomagnetic sensing scales
picotesla	$10^{-12}$ T	biomagnetism and precision laboratory signals
nanotesla	$10^{-9}$ T	small anomalies / precision field changes
microtesla	$10^{-6}$ T	geomagnetic-scale quantities
Earth’s field	roughly $25$–$65\,\mu$T	location dependent
millitesla	$10^{-3}$ T	laboratory bias magnets, permanent magnets at distance
tesla	1 T	strong laboratory magnets
clinical MRI	commonly 1.5–3 T	main $B_0$ field
high-field MRI/research	7 T and above	specialized systems

For ground-state atomic magnetometry the meaningful scale is often not only $B$, but $\gamma B$ relative to linewidth and relaxation rates.

Electric-field scales

Electric field spans an enormous range. A useful relation for an ideal free-space plane wave is

\[\langle S\rangle=\frac{E_{\rm rms}^2}{\eta_0}.\]

Therefore

\[E_{\rm rms}=\sqrt{\eta_0\langle S\rangle}.\]

For example, a plane-wave power density of $1\ \mathrm{W/m^2}$ corresponds to approximately $19.4\ \mathrm{V/m}$ rms.

AssumptionThis relation is a far-field plane-wave relation. It must not be applied blindly in reactive near fields, waveguides, resonators or strongly inhomogeneous structures.

Copper skin depth

For a good conductor,

\[\delta=\sqrt{\frac{1}{\pi f\mu\sigma}}.\]

For copper with $\sigma\approx5.8\times10^7\ \mathrm{S/m}$ and $\mu_r\approx1$:

Frequency	Skin depth
60 Hz	≈ 8.5 mm
1 MHz	≈ 66 μm
1 GHz	≈ 2.1 μm
10 GHz	≈ 0.66 μm

The $1/\sqrt f$ dependence is more important than memorizing any one value.

Time scales in RF and atomic experiments

Time	Corresponding inverse frequency	Typical interpretation
1 s	1 Hz	slow drift / field changes
1 ms	1 kHz	modulation / slow control
1 μs	1 MHz	atomic/RF transients
1 ns	1 GHz	microwave cycle
1 ps	1 THz	ultrafast / THz cycle
1 fs	1 PHz	optical-cycle scale

A system cannot follow modulation much faster than the inverse of its relevant response time or bandwidth.

Antenna size and far field

For maximum antenna dimension $D$,

\[R_{FF}\gtrsim\frac{2D^2}{\lambda}\]

is a common far-field criterion.

This can be surprisingly large for electrically large apertures. A large phased array can require a far-field range much longer than “a few wavelengths.”

Atomic and Rydberg scales

Useful mental hierarchy:

ground-state atomic dimensions: ångström scale;
highly excited Rydberg electronic radius: grows approximately as $n^2$;
optical transition frequencies: hundreds of THz;
microwave Rydberg transitions: often MHz–100s of GHz depending on states;
EIT/atomic-resonance linewidths: experiment dependent, often kHz–MHz scales;
detector/readout bandwidth can be far smaller than optical frequency because only the envelope or beat signal is measured.

Common misconceptionA 500-THz optical carrier does not require a 500-THz photodetector to measure slowly varying optical transmission. The detector follows the intensity envelope or heterodyne beat within its electrical bandwidth.

Build your own scale estimate

Before a detailed calculation, write three lines:

Characteristic length $L$ and wavelength $\lambda$.
Characteristic time $\tau$ and inverse frequency $1/\tau$.
Characteristic field/power and the expected detector/noise scale.

Then ask whether your exact answer is consistent with all three.