12. Electromagnetic wavesSavchenko Formulas, chapter 12 of 14, 10 formulas
Sections follow the book, and within a section the most used formulas come first. An italic problem number means the formula appears in its statement. Rest the cursor on a number to see the statement.
$$E = E_0\sin(\omega t - kz), \qquad k = \frac{2\pi}{\lambda} = \frac{\omega}{c}, \qquad \lambda\nu = c$$
$\displaystyle \frac{\partial^2 E}{\partial z^2} = \frac{1}{c^2}\frac{\partial^2 E}{\partial t^2}\ \text{(волновое уравнение)}$$\displaystyle E = 2E_0\sin kx\,\sin\omega t\ \text{(стоячая волна у зеркала)}$
E0
field amplitude
$\omega, k$
angular frequency and wave number
$\lambda$
wavelength
c
speed of light
The phase $\omega t - kz$ is constant on a plane travelling at $\omega/k = c$, so wavelength and frequency are tied by $\lambda\nu = c$, and the phase difference between two points along a ray is $2\pi d/\lambda$. From Maxwell's equations without charges follows the wave equation, satisfied by any function of $t - z/c$. Two waves of one frequency add as vectors by phase, and equal counter-propagating waves make a standing wave with nodes every half wavelength.
$$E = cB, \qquad c = \frac{1}{\sqrt{\varepsilon_0\mu_0}}, \qquad v = \frac{c}{\sqrt{\varepsilon\mu}} = \frac{c}{n}$$
$\displaystyle \frac{\varepsilon_0 E^2}{2} = \frac{B^2}{2\mu_0}\ \text{(энергия поровну между полями)}$$\displaystyle n = \sqrt{\varepsilon\mu}\ \text{(показатель преломления)}$
E, B
amplitudes of the wave's electric and magnetic fields
$\varepsilon_0, \mu_0$
electric and magnetic constants
$\varepsilon, \mu$
permittivity and permeability of the medium
Applying Faraday's law to a loop that the wave front crosses at speed $c$ gives $E = cB$, and the circulation theorem with displacement current gives $B = \varepsilon_0\mu_0 cE$, whence $c = 1/\sqrt{\varepsilon_0\mu_0}$. The fields are perpendicular to each other and to the direction of travel and carry equal energies. In a medium the speed is $\sqrt{\varepsilon\mu}$ times smaller, which is the refractive index.
$\displaystyle i = \frac{2E}{\mu_0 c}\ \text{(ток в зеркале, гасящий волну за ним)}$$\displaystyle R = \left(\frac{n_e e^2 x}{4\pi\varepsilon_0 m_e c\nu}\right)^2\ \text{(отражение тонкой плёнки плазмы)}$
i
surface current density of the radiating sheet
$\sigma, v$
charge density of the sheet and its speed
E
field of the sheet at rest, $\sigma/2\varepsilon_0$
x
distance from the sheet
A sheet with a changing current radiates on both sides a wave with magnetic field $\mu_0 i/2$ and electric field $cB$, arriving at a point $x$ with delay $x/c$. For a charged sheet given a jolt the radiated field is the static field times $v/c$, pointing against the velocity and weakening with distance only through the delay. A perfect mirror sets up the current $2E/\mu_0 c$, whose wave cancels the incident one behind it, and a thin plasma film reflects a fraction growing with electron density and falling with frequency.
$$\vec S = \frac{[\vec E\times\vec B]}{\mu_0}, \qquad I = \langle S\rangle = \frac{\varepsilon_0 c E_0^2}{2}, \qquad w = \varepsilon_0 E^2$$
$\displaystyle E \propto \frac{1}{r}, \qquad I \propto \frac{1}{r^2}\ \text{(сферическая волна)}$$\displaystyle E_0 = \sqrt{\frac{2I}{\varepsilon_0 c}}$
$\vec S$
Poynting vector, energy flow per unit area per second
I
intensity, the mean energy flux
w
energy density of the wave, $\varepsilon_0E^2/2 + B^2/2\mu_0$
The wave's energy density is $\varepsilon_0E^2$, half in the electric and half in the magnetic field, and it moves at $c$, hence the flux $\varepsilon_0cE^2$ and the mean $\varepsilon_0cE_0^2/2$. Intensity goes as the square of the amplitude, so in a spherical wave the amplitude falls as $1/r$ while the power through any sphere stays the same. From a source's power one finds the field at a given distance.
$\displaystyle F = \frac{P}{c}\ \text{(сила на чёрную пластинку, поглощающую мощность }P)$$\displaystyle p = \frac{\varepsilon_0 E^2}{2}\,\frac{c + v}{c - v}\ \text{(зеркало, летящее навстречу)}$
I
intensity of the wave
$\rho$
reflectivity
$\theta$
angle of incidence
E0
amplitude of the incident wave
A wave carries momentum $w/c$ per unit volume, and an absorbing wall feels a pressure $I/c$, a mirror twice that since the reflected wave carries momentum back. At oblique incidence a factor $\cos^2\theta$ appears, one cosine from projecting the momentum, the other from the area. The same pressure comes out as the Lorentz force on the currents induced in the mirror, $\varepsilon_0E^2$ for the standing wave at a mirror. A mirror moving toward the wave feels more pressure through the Doppler rise of frequency.
$\displaystyle d = (2k + 1)\frac{\lambda}{4}\ \text{(узел у одного зеркала, пучность у другого)}$$\displaystyle \Delta = 2d = k\lambda\ \text{(интерференция при отражении от двух плоскостей)}$
d
distance between the mirrors or thickness of the layer
$\lambda$
wavelength
k
an integer
Between two conducting planes the field vanishes at each, so a whole number of half waves fits and the resonant frequencies are spaced by $c/2d$. With one reflecting plane and an open end there is a node at one edge and an antinode at the other, and an odd number of quarter waves fits. The same count of half waves gives the conditions for reinforcement and cancellation of waves reflected from two parallel planes and the tuning of antennas.
The frequency of light depends only on the relative speed of source and receiver, and the exact formula follows from the Lorentz transformation of the wave's phase or of the fields, the field in the moving frame being $\sqrt{(1+v/c)/(1-v/c)}$ times larger. At low speeds the frequency shift is $\nu_0 v/c$, doubled on reflection from a moving mirror, which is how radar works and how stellar velocities are read from line shifts.
angles of incidence and refraction, from the normal
v1, v2
wave speeds in the two media
n
refractive index, $c/v$
Every point of the front is a source of secondary waves, and in a time $\tau$ the front advances $v_1\tau$ in the first medium and $v_2\tau$ in the second, hence the law of reflection and sines of angles in the ratio of the speeds. The frequency is unchanged on entering a medium while the wavelength shrinks $n$ times. If a source outruns the waves the envelope of secondary waves forms a cone with $\cos\theta = v_{\text{wave}}/v$, Cherenkov radiation or a shock wave.
$\displaystyle l \approx \frac{r\lambda}{D}\ \text{(размер пятна на расстоянии }r)$$\displaystyle I_{\text{расс}} \propto \frac{1}{\lambda^4}\ \text{(рассеяние Рэлея на мелких частицах)}$
b
width of the slit
D
size of the aperture, antenna, mirror
$\theta$
divergence angle of the beam
d
grating period or distance between sources
A wave through an aperture of width $b$ spreads by an angle of order $\lambda/b$, since the secondary waves from the slit's edges cancel when their path difference equals a wavelength. So an antenna or mirror of size $D$ cannot make a beam narrower than $\lambda/D$ nor resolve details closer than that angle, and the spot at distance $r$ has size $r\lambda/D$. Two sources or a grating reinforce the wave in directions where the path difference is a multiple of $\lambda$. Small particles scatter short waves more strongly, as $1/\lambda^4$.
$\displaystyle r_1 = \sqrt{\lambda L}\ \text{(первая зона на расстоянии }L\text{ от точки)}$$\displaystyle l_m = \frac{r^2}{(2m - 1)\lambda}\ \text{(расстояния, на которых отверстие радиуса }r\text{ открывает }m\text{ зон)}$
a
distance from the source to the aperture
b
distance from the aperture to the observation point
rm
radius of the $m$th zone
Am
amplitude from the $m$th zone, all nearly equal
The wave front is cut into rings whose neighbours reach the point with a phase shift of $\pi$, their contributions alternate in sign and are nearly equal in size, so the whole open wave gives half the first zone's contribution. The zones have equal areas, $\pi\lambda ab/(a+b)$. An aperture opening an odd number of zones gives a maximum at the centre, an even one a minimum, and moving the observation point along the axis gives alternating maxima at distances $r^2/(2m-1)\lambda$. A disc covering a few zones leaves a bright spot at the centre of its shadow.