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Electromagnetic Waves

Saturday, January 22, 2022

Review of Electromagnetism

Electric charges produce an electric field, E\boldsymbol{\overrightarrow{E}}. Flowing current in a wire produces a magnetic field, B\boldsymbol{\overrightarrow{B}}. If charges do not move and current does not flow, the electric and magnetic fields are said to be static.

Accelerating charges and currents that vary with time produce electromagnetic waves, where the E\boldsymbol{E} and B\boldsymbol{B} fields vary over time.

Electromagnetic Wave Basics

Plane waves

Plane waves (when wave fronts are planes) are the simplest versions of electromagnetic waves. They can be described as such:

E=E0sin(kzωt)B=B0sin(kzωt)\boldsymbol{\overrightarrow{E}}=\boldsymbol{\overrightarrow{E_0}}\sin{\left(kz-\omega t\right)} \newline \boldsymbol{\overrightarrow{B}}=\boldsymbol{\overrightarrow{B_0}}\sin{\left(kz-\omega t\right)}

Where kk is the wave number, found from the wavelength: k=2π/λk=2\pi / \lambda, and ω\omega is the angular frequency, found from the frequency: ω=2πf\omega = 2\pi f.

Since c=λfc=\lambda f, the wave number and angular frequency are related as such:



Polarization is determined by the vector E0\boldsymbol{\overrightarrow{E_0}} and the direction of propagation of the electromagnetic wave. The direction of B0\boldsymbol{\overrightarrow{B_0}} must be perpendicular to both E\boldsymbol{\overrightarrow{E}} and the direction the wave propagates, such that E×B\boldsymbol{\overrightarrow{E}} \times \boldsymbol{\overrightarrow{B}} is the direction of the wave.

E0E_0 and B0B_0 are related such that B0=E0/cB_0 = E_0 / c.

Poynting vector, S\boldsymbol{\overrightarrow{S}}

The electromagnetic wave transmits energy according to its Poynting vector, which is defined as such:

S=1μ0E×B\boldsymbol{\overrightarrow{S}} = \frac{1}{\mu_0}\boldsymbol{\overrightarrow{E}}\times \boldsymbol{\overrightarrow{B}}

Where μ0\mu_0 is the vacuum permeability constant, approximately 4π107H/m4\pi * 10^{-7} H/m.

For the plane wave, the Poynting vector is determined as so:

S=1μ0E0B0sin2(kzωt)k^\boldsymbol{\overrightarrow{S}}=\frac{1}{\mu_0}E_0B_0\sin^2{\left(kz-\omega t\right)}\boldsymbol{\hat{k}}

And since the power received is equal to the energy times area, the power delivered by a plane wave is determined like this:

P=SA=1μ0E0B0Asin2(kzωt)P=SA=\frac{1}{\mu_0}E_0B_0A\sin^2{\left(kz-\omega t\right)}

Or, since B0=E0/cB_0=E_0/c,

P=1μ0cE02Asin2(kzωt)P=\frac{1}{\mu_0c}E_0^2A\sin^2{\left(kz-\omega t\right)}

Notice how the power delivered is proportional to E02E_0^2. This means the intensity, average power per unit area, is proportional to the square of the amplitude of the wave.

Average power

According to the equation above, intensity fluctuates with a frequency of 2f=2(ω/2π)2f=2\left(\omega/2\pi\right).

Sensors (like our eyes) can't see fluctuations in intensity on the order of 101510^{15} times per second. Therefore, we can find the average power as such:

Paverage=1T0TP dtP_{average}=\frac{1}{T}\int_0^TP~dt


Waves Interacting

Principle of superposition

Waves can interact with each other through the processes of interference and diffraction, because of the principle of superposition, which states that waves can "add together" their amplitudes at points.

Double-slit experiment

A plane wave approaches a wall with two slits in it. On the other side of the wall, the wave is diffracted, such that it covers a greater area than it otherwise would. This causes the waves to interfere with each other, producing a pattern of lines on the screen.

The brighter spots are caused by constructive interference, which is when two waves' crests intersect. The dimmer spots are caused by destructive interference, which is when one wave's crest intersects another wave's trough (valley).

Diffraction Grating

Diffraction grating allows light waves to be split into their constituent wavelengths, according to the following equation:


Where dd is the slit spacing and nn is the order of diffraction (an integer), which represents which set of wavelengths of light the angle refers to (see image below).


Diffraction grating is useful if dd is only a few times the wavelength, making it impossible for short-wavelength light like x-rays.

X-ray crystal diffraction

For short wavelength light, crystalline structures of materials are used to diffract the light. Interference maxima occur according to Bragg's Law of X-ray Diffraction:


X-ray crystal diffraction