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Electromagnetic Wave: Fields E and B, Polarization and Polarizers

Light, radio, Wi-Fi and X-rays are one and the same wave: two fields, E and B, perpendicular to each other, holding on to one another and running together at the speed c, carrying energy. Drag the frequency along the spectrum, change the polarization, put polarizers in the way — and watch what happens to the fields, to a charge in the path of the wave and to the energy flow.
All numbers in the panels are real, in SI; on the scene the wave is slowed down and stretched so that the eye can follow it.

Drag to rotate, scroll to zoom

E(z) and S(z) right now: a “snapshot” of the wave along the axis

Polarization figure: what the tip of E draws

What is what on the scene

Red arrows — the electric field E: at each point of the axis the arrow shows where and how hard the field would push a positive charge. It sticks out perpendicular to the axis and all the tips lie on a glowing curve.
Blue arrows — the magnetic field B: always at a right angle to E and to the axis. It is the same red arrow turned by 90°: B = E/c. On the scene they have equal length, while in real units the number B is 3·10⁸ times smaller than the number E.
Red and blue “sheets”: the trace of the arrows, so that you can see the whole shape of the wave. For a linear wave the sheet is flat, for a circular one it is twisted like a screw, for an elliptical one — a flattened screw.
Green sparks on the axis — the energy flow S: the stronger the field at that place, the bigger the spark (S ∝ E²). It is never negative: the wave carries energy only forward, although E and B swing back and forth.
Yellow ring — the chosen point z: E and B are drawn large there, and the graphs and the “Measurements” tab show their numbers. Move the point with the z slider to compare the place behind a polarizer with the one in front of it.
Golden ball — a test charge: an electron on a “spring” (as in an atom or an antenna). The field E sways it along itself — this is how a radio wave induces a current in an antenna and light does so in the retina.
Ring on the left — the source: the red “+” and the blue “−” swing together with the field: a wave is born from oscillating charges. For visible light the ring glows in the colour of the wave.
Coloured discs P₁, P₂, P₃ — polarizers: the strokes and the thick bar show the transmission axis α: only the component of E along it passes, the transverse one is absorbed. Behind the disc the wave is linearly polarized along α.

f — frequency

Drag along the scale or click: from radio (left) to X-rays (right). The rainbow “magnifier” enlarges visible light.

Polarization of the wave

ψ — polarization angle 45°

E_y/E_x — ratio of swings 0.50

δ — phase shift 60°

Polarizers (tick P)

P₁ 0°

P₂ 45°

P₃ 90°

E₀ — field amplitude 50

z — observation point 2.0

Playback speed (slow motion) 0.5×

Show

Electric field E

Magnetic field B

Energy flow S (green sparks)

Translucent surfaces

Glowing curves through the tips

Test charge

Gentle camera sway

The scale on the scene is conventional: 10¹⁴ oscillations per second cannot be seen, so the wave is slowed down and λ is drawn so that several crests fit. All numbers in the panels are real, in SI. Right/left handedness follows the optical rule: we look facing the wave.

The idea in 30 seconds

A wave is two fields holding on to each other

A changing magnetic field creates an electric one, and a changing electric field creates a magnetic one. Propping each other up, they run by themselves, with no medium and no wires, at the speed c = 3·10⁸ m/s. E and B are always perpendicular to each other and to the direction of travel, they rise and fall together, and their ratio is c: E = c·B.

The field sways back and forth, the energy goes only forward

The Poynting vector S = E×B/μ₀ shows where the energy flows. During the oscillation the signs of E and B change, but E×B does not: S always looks forward (the green sparks). The average power per unit area is the intensity I = ½ε₀cE₀². Double the amplitude and the intensity grows four times.

Polarization is the “drawing” of the tip of E

Looking toward the wave, the tip of E moves along a straight line (linear), a circle (circular) or an ellipse. Two numbers define it: the ratio of the swings along x and y and the phase shift δ. A polarizer is a “slit”: it lets through only the component along its axis, so I = I₀cos²Δ (Malus’s law).

Radio and light are the same thing

They differ only in frequency, and therefore in wavelength λ = c/f: FM radio — 3 m, Wi-Fi — 12 cm, red light — 700 nm, X-rays — 0.3 nm. Drag f along the spectrum: the structure of the wave stays the same, only the “density” of the crests changes.

Try it yourself

Press “Red light”, then “Violet”: the crests get denser — the frequency grows and the wavelength falls (λ = c/f), while the speed stays the same. Then “X-rays”: the wave is thousands of times shorter.
Choose “Circular, left-handed” and press “Along the wave”: the tip of E runs round a circle and the red “sheet” in 3D is twisted like a screw. Switch left to right and the rotation reverses.
Press “One: α = 45°” (the wave is linear along x): the intensity drops exactly by half — cos²45° = ½. Turn the axis of P₁: at 0° everything passes, at 90° nothing does.
Press “Two crossed”: the light is extinguished. Now “Three: 0°, 45°, 90°” — and a quarter of the light is back, although we added one more filter! The middle plate “turns” the field, and the last plate now has something to pass.
Move the point z and watch the graph: the red line of E and the blue underlay c·B lie exactly on top of each other — the fields are in phase, their zeros and maxima coincide. And the green graph of S never goes below zero.
Watch the golden ball: the larger E₀, the harder it is shaken. An antenna and an eye work the same way — the field shakes charges and the receiver picks up this response.

Where it works in real life

Radio, Wi-Fi and the phoneAn antenna is a piece of wire in which the field of the wave swings the electrons. The receiver only has to amplify the tiny current. FM radio (3 m) and Wi-Fi (12 cm) differ only in frequency.

Microwave ovenA wave with f ≈ 2.45 GHz quickly reverses the field E inside the food. Water molecules rotate after it and heat their neighbours.

Polarized sunglassesLight reflected from water and asphalt is polarized horizontally. The glasses pass only the vertical component and cut the glare — this is the very same polarizer.

LCD screenThe backlight passes through two crossed polarizers, and the liquid crystal between them rotates the polarization in every pixel: the larger the rotation, the brighter the dot.

The eye, warmth and a tanThe eye catches waves with λ from 380 to 750 nm: red is about 430 THz, violet about 750 THz. A bit longer than red is infrared heat, a bit shorter than violet is the ultraviolet that gives a tan.

X-raysThe same wave, but f ≈ 10¹⁸ Hz (λ ≈ 0.3 nm): it passes freely through soft tissue and is stopped by bones — hence the shadows on the image.

The frequency f and the wavelength λ are tied together by the speed of light: λ = c/f. So from radio waves (metres) to X-rays (tenths of a nanometre) it is enough to drag a single slider f. The wavelength on the scene is conventional, so that several crests fit, while the “Measurements” panel shows honest values: c = 3·10⁸ m/s, B₀ = E₀/c, I = ½ε₀cE₀² = ⟨S⟩.

What to explore next

Vortex electric fieldWhere E comes from when B changes — the first half of the wave mechanism. Snell’s law and total internal reflectionWhat happens to a light wave at the boundary of two media. Wave interferenceHow two waves add up, strengthening or cancelling each other.

KEY FORMULAS

Plane electromagnetic wave

$$E_x = E_0 \cos(kz - \omega t), \qquad B_y = \frac{E_x}{c} = \frac{E_0}{c}\cos(kz - \omega t)$$

The same sine wave runs along +z: E and B are in phase, perpendicular to each other and to the z axis, |B| = |E|/c

▶ Detailed Explanation

A monochromatic plane wave in vacuum runs along +z with the wave number k = 2π/λ and the angular frequency ω = 2πf. The expression kz − ωt is called the phase: a point of constant phase (a crest, say) moves with the speed ω/k = c. If E points along x, then B points along y: B = (1/c)·ẑ × E, so |B| = |E|/c, and E and B reach their maxima and zeros at the same moments — on the graph the red line lies exactly on the blue one. On the scene E and B are drawn with equal length (a conventional scale), while the numbers in the panels are real.

Speed of light, frequency and wavelength

$$c = \lambda f, \qquad \omega = 2\pi f, \qquad k = \frac{2\pi}{\lambda} = \frac{\omega}{c}$$

In vacuum the speed does not depend on the frequency: λ and f are inversely related through c = 3·10⁸ m/s

▶ Detailed Explanation

The speed of light in vacuum is a universal constant c ≈ 3·10⁸ m/s; it depends neither on the frequency nor on the amplitude. Hence λ = c/f: double the frequency and the wave becomes twice as short. The same quantities in angular form: ω = ck. For reference: FM radio, f = 100 MHz — λ = 3 m; Wi-Fi, 2.4 GHz — λ = 12.5 cm; green light, 550 nm — f = 5.45·10¹⁴ Hz; X-rays, f = 10¹⁸ Hz — λ = 0.3 nm. In the simulator f changes on a logarithmic scale from 10⁶ to 10¹⁸ Hz and λ is recalculated automatically; the visible range (380–750 nm) is only a narrow strip, and the “magnifier” on the scale enlarges it.

The relation E = cB and the direction k

$$E = cB, \qquad \vec{E} \perp \vec{B} \perp \vec{k}, \qquad \vec{E} \times \vec{B} \parallel \vec{k}$$

E and B are perpendicular to each other and to the direction of propagation, and their magnitudes are tied through c

▶ Detailed Explanation

For a wave in vacuum the vectors E, B and k form a right-handed triple: E × B points along the energy flow. Right-hand rule: from E to B, the thumb shows where the wave runs. The magnitudes are tied by E = cB: for E₀ = 50 V/m we get B₀ = E₀/c ≈ 167 nT — a number 3·10⁸ times smaller, although the electric and magnetic fields carry equal energy (the densities ε₀E²/2 and B²/(2μ₀) are equal). That is why on the scene both arrows have the same length, and the honest ratio |E|/|B| = c is shown in the “Measurements” panel — it is computed from the real numbers at the chosen point.

Poynting vector

$$\vec{S} = \frac{1}{\mu_0}\,\vec{E} \times \vec{B}, \qquad S = \frac{EB}{\mu_0} = \varepsilon_0 c E^2$$

Energy flux density: directed along E × B, its magnitude is proportional to the square of the instantaneous field

▶ Detailed Explanation

The Poynting vector S = E × B / μ₀ is the energy flux density: its magnitude in watts per square metre shows how much energy the wave carries through a unit area at the given moment. The direction of S coincides with k: take E along x and B along y — and S is along z. Although E and B oscillate and change sign, their product is positive: S = ε₀cE² ≥ 0, the wave carries energy only forward. On the scene S is the green sparks on the axis; their size is proportional to E² and they pulse twice as often as the field itself. The value of S averaged over a period is the intensity.

Intensity of a monochromatic wave

$$I = \langle S \rangle = \frac{E_0 B_0}{2\mu_0} = \frac{1}{2}\varepsilon_0 c E_0^2$$

The power per unit area averaged over a period — depends only on the amplitude E₀

▶ Detailed Explanation

The instantaneous S oscillates, so in practice one takes the average over a period — the intensity. The average of cos² is ½, hence I = ½ε₀cE₀² = E₀B₀/(2μ₀). The intensity grows as the square of the amplitude: double E₀ and there is four times more energy. For E₀ = 50 V/m we get I ≈ 3.3 W/m² (for comparison, sunlight at the Earth’s surface is about 1000 W/m², which corresponds to E₀ ≈ 870 V/m). In the simulator the amplitude E₀ is the same for any polarization: circular and linear waves with equal E₀ carry the same intensity, and behind a polarizer it drops — you can see it in the HUD and in “Measurements”.

Polarization: linear, circular, elliptical

$$E_x = E_{0x}\cos\theta, \qquad E_y = E_{0y}\cos(\theta + \delta), \qquad \theta = kz - \omega t$$

The type of polarization is set by the ratio of the amplitudes of the components and the phase shift δ between them

▶ Detailed Explanation

In the xy plane the tip of the vector E draws different curves depending on two numbers — the ratio E₀y/E₀x and the phase shift δ. If δ = 0 or π (or one of the components is zero), the path is a segment: this is linear polarization with the plane angle ψ = arctan(E₀y/E₀x). Equal amplitudes and δ = ±90° give a circle — circular polarization. Everything else is an ellipse. The sense of rotation is determined by looking toward the wave: here the optical rule is used — counter-clockwise is “left”, clockwise is “right” (radio engineering uses the opposite). Check it: choose circular, press “Along the wave” and look at the polarization figure: the yellow comet runs round a circle, and the red “sheet” in 3D is twisted like a screw.

Malus’s law: the polarizer

$$E' = E_0\cos\Delta, \qquad I = I_0\cos^2\Delta, \qquad \Delta = \psi - \alpha$$

A polarizer passes only the component of E along its axis: the intensity falls as the cos² of the angle between the polarization and the axis

▶ Detailed Explanation

A polarizer is a “slit” with the transmission axis α: it leaves the vector E only its projection on that axis, and the transverse component is absorbed. If the wave is linearly polarized at the angle ψ, the amplitude behind the plate is E′ = E₀cosΔ, where Δ = ψ − α, and since I ∝ E², we get I = I₀cos²Δ. At Δ = 0 everything passes, at Δ = 45° half, at Δ = 90° nothing. For a circular wave the projection on any axis gives exactly ½I₀, and behind the plate the wave becomes linear along α. The simulator computes this exactly: behind each plate the field is the projection of the complex amplitude on the axis, and in “Measurements” the figure is checked against cos²Δ. Natural light (a lamp, the Sun) is a mixture of all ψ, so the first polarizer passes exactly ½.

Three polarizers: the quarter paradox

$$I = I_0\cos^2\Delta_1\,\cos^2\Delta_2, \qquad \Delta_1 = \Delta_2 = 45^\circ \;\Rightarrow\; I = \frac{I_0}{4}$$

Every plate is a projection; inserting a third one at 45° between crossed plates brings back a quarter of the light

▶ Detailed Explanation

Take a linear wave along x (ψ = 0°). Plate P₁ with the axis 0° passes it entirely, and P₃ with the axis 90° is crossed with it and extinguishes everything: Δ = 90°, cos²90° = 0. Now put P₂ with the axis 45° between them. After P₂ the wave is linear at 45°, its amplitude is E₀cos45° and its intensity I₀/2. For P₃ the angle between the polarization and the axis is again 45°, so another half passes: I = I₀·½·½ = I₀/4. This does not violate energy conservation — plates only absorb and never add light; the crossed pair passed zero simply because the field was strictly perpendicular to the axis, and the middle plate gave it the necessary component. In the simulator press “Three: 0°, 45°, 90°” and compare with “Two crossed”.

An electromagnetic wave is a disturbance of the electric and magnetic fields that travels by itself, with no medium and no wires. A changing magnetic field creates a vortex electric field, and a changing electric field creates a magnetic one; propping each other up, the fields fly through vacuum at c = 3·10⁸ m/s. At every point the vectors E and B oscillate perpendicular to each other and to the direction of propagation k, and their magnitudes are tied by E = cB. That is why on the scene the blue arrow B is the same as the red E turned by 90° (in real units, for E₀ = 50 V/m one gets B₀ ≈ 167 nT).

The Poynting vector S = E × B / μ₀ shows where the energy goes. Although E and B change sign now and then, their cross product always points forward: S = ε₀cE² ≥ 0 — these are the green sparks on the axis, and the stronger the field, the bigger they are. The average of S is the intensity I = ½ε₀cE₀², the power the wave brings to each square metre. It grows as the square of the amplitude: double E₀ and there is four times more energy.

From radio to X-rays it is the same wave; only the frequency f and the wavelength λ = c/f differ: FM radio — about 3 m, Wi-Fi — 12 cm, red light — 700 nm, X-rays — tenths of a nanometre. The slider f spans 12 orders of magnitude (10⁶…10¹⁸ Hz), while visible light is only a narrow strip between 380 and 750 nm, so the spectrum scale has a “magnifier”.

Polarization describes what the tip of the vector E draws when you look toward the wave: a segment (linear), a circle (circular) or an ellipse. A circle appears when the swings along x and y are equal and the phase shift is ±90°. The sense of rotation is called left or right; here the optical rule is used — counter-clockwise when facing the wave is “left” (in radio engineering the convention is the opposite).

A polarizer lets through only the component of E along its axis α. If the wave is linearly polarized at the angle ψ, the amplitude behind the plate is E₀cosΔ and the intensity is I = I₀cos²Δ, where Δ = ψ − α — this is Malus’s law. Natural light (a lamp, the Sun) is a mixture of all directions, and the first polarizer takes exactly half. Two crossed plates (Δ = 90°) extinguish the light completely, but if a third one at 45° is inserted between them, ¼ of the light gets through: the middle plate has “turned” the polarization and the last one again has something to pass. Polarized sunglasses and liquid-crystal screens are built on this.

How to use it: press “Red light”, then “Violet” and “X-rays” — you will see the crests crowd together. Choose circular polarization and look “Along the wave”. Tick P₁…P₃ and build the experiment with three polarizers. Move the point z and compare the numbers in “Measurements” with the formulas: |E|/|B| always equals c, and the intensity behind the plates agrees with Malus’s law.