Electromagnetic Waves Calculator: Wavelength and Optics

One speed, and everything else follows from it

Maxwell's equations predict waves travelling at 1/√(ε₀μ₀), and that number coincides with the experimentally measured speed of light. It was the discovery that unified optics and electromagnetism: light is not a phenomenon of its own, it is an electric field and a magnetic field sustaining each other as they travel. Radio, microwaves, infrared, light, ultraviolet, X-rays and gamma rays are the same thing at different frequencies.

Because the speed is fixed, wavelength and frequency are tied by c = λf and cannot be chosen independently: saying "500 nanometres" and saying "600 terahertz" is saying the same thing. In a medium the speed drops to c/n, and the wavelength with it, while the frequency stays that of the source. It is precisely this change of speed that bends rays at an interface, and that is the content of Snell's law.

Interference is what demonstrates the wave nature of light. Two close slits produce bright and dark fringes on a screen: where the two paths differ by a whole number of wavelengths the crests add, where they differ by half a wavelength they cancel. No particle model explains why opening a second slit can make darker a point that was lit before.

Common mistakes

Using the vacuum wavelength for calculations inside a medium: on entering water the wavelength shrinks by the factor n, while the frequency does not change.
Measuring Snell angles from the surface instead of from the normal: a grazing ray has an angle of 90°, not 0°.
Forgetting that beyond the critical angle no refracted ray exists: if the required sine exceeds 1 the answer is not an arithmetic slip, it is total internal reflection.

Frequently asked questions

How do you convert a wavelength into a frequency?

With f = c/λ, where c = 299,792,458 m/s. Green light at 500 nanometres corresponds to about 600 terahertz; an FM radio wave of 3 metres corresponds to about 100 megahertz.

What is the formula for Snell's law?

n₁·sin θ₁ = n₂·sin θ₂, with the angles measured from the normal to the surface. Entering a medium of higher index bends the ray towards the normal.

What is the critical angle, and when does total internal reflection happen?

It is the angle of incidence beyond which a ray going from a denser medium to a thinner one can no longer get out: sin θ_c = n₂/n₁. Past that angle the light is completely reflected, which is the principle of optical fibre.

How much energy does a photon of visible light carry?

Between about 1.8 and 3.3 electronvolts, from E = h·f. Red at 700 nm carries 1.77 eV, violet at 400 nm carries 3.10: that is why ultraviolet damages molecules and infrared does not.

How this calculation works

Fundamental relation: c = λ·f, with c = 299,792,458 m/s in vacuum. In a medium of index n: v = c/n and λ_medium = λ_vacuum/n, while the frequency is unchanged. Photon energy: E = h·f = h·c/λ, with h = 6.626 × 10⁻³⁴ J·s; 1 eV = 1.602 × 10⁻¹⁹ J. Refractive index: n = c/v. Snell's law: n₁·sin θ₁ = n₂·sin θ₂. Critical angle: sin θ_c = n₂/n₁, defined only when n₁ > n₂. Intensity of a plane wave: I = E₀²/(2·μ₀·c), with B₀ = E₀/c. Double-slit interference: maxima at d·sin θ = m·λ, minima at d·sin θ = (m + ½)·λ. Fringe spacing on a screen at distance D: Δy ≈ λ·D/d.

Mechanical waves

String speed, sound in air, Doppler, beats and sound level.

Polarisation and diffraction

Wave optics: Malus, Brewster, diffraction, resolving power and interference.

Lenses and mirrors

Geometrical optics: focal length, image distance, magnification and optical power.

Refraction and Snell's law

Angle of refraction, critical angle and total internal reflection, with a draggable ray diagram.