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 Maxwell's Equations with No Charges
 Wave Equations
 Solution of Wave Equation
 Electromagnetic Spectrum
 Accelerated Charges
 Why is the Sky Blue
 Source and Reference

Maxwell's Equations with No Charges

In the absence of sources, charges, current Gauss's Law: 𝐸=1𝜀0𝜌⇒𝐸=0
Gauss's Law (Magnetism): 𝐵=0 ⇒𝐵=0
Faraday's Law: ×𝐸=−𝐵∂𝑡×𝐸=−𝐵∂𝑡
Ampere-Maxwell Law: ×𝐵=𝜇0𝐽+𝜀0𝐸∂𝑡×𝐵=𝜇0𝜀0𝐸∂𝑡.

Wave Equations

Once a wave starts, Both Faraday and Ampere equations keep going. Feed one equation into the other: ××𝐵=𝜇0𝜀0𝐸∂𝑡
××𝐵=𝜇0𝜀0∂𝑡×𝐸=𝜇0𝜀0∂𝑡𝐵∂𝑡=−1𝑐22∂𝑡2𝐵
⇒−2∂𝑥2+2∂𝑦2+2∂𝑧2𝐵=−1𝑐22∂𝑡2𝐵
2∂𝑥2+2∂𝑦2+2∂𝑧2𝐵=1𝑐22∂𝑡2𝐵
and similarly
2∂𝑥2+2∂𝑦2+2∂𝑧2𝐸=1𝑐22∂𝑡2𝐸
In other words, both 𝐸 and 𝐵 are waving.

Solution of Wave Equation

Let 𝐸 be wave of the form: 𝐸=𝐸0sin (𝑘𝑥−𝜔𝑡)𝑧
2∂𝑥2+2∂𝑦2+2∂𝑧2𝐸=2∂𝑥2𝐸=−𝑘2𝐸0sin (𝑘𝑥−𝜔𝑡)𝑧
1𝑐22∂𝑡2𝐸=−𝜔2𝑐2𝐸0sin (𝑘𝑥−𝜔𝑡)𝑧
If 𝜔=𝑐𝑘 then left and right terms are the same. And 𝐸=𝐸0sin (𝑘𝑥−𝜔𝑡)𝑧 is the wave solution of 𝐸.
For the differential form of Faraday's law, ×𝐸=−𝐵∂𝑡, let 𝐸 be wave of the form: 𝐸=𝐸0sin (𝑘𝑥−𝜔𝑡)𝑧, then
×𝐸=𝑥𝑦𝑧∂𝑥∂𝑦∂𝑧00𝐸𝑧=∂𝑦𝐸𝑧𝑥∂𝑥𝐸𝑧𝑦=−𝑘𝐸0cos (𝑘𝑥−𝜔𝑡)𝑦≡−𝐵∂𝑡
𝐵=𝐸0𝑘𝜔sin (𝑘𝑥−𝜔𝑡)𝑦=𝐸0𝑐sin (𝑘𝑥−𝜔𝑡)𝑦
Self-sustaining wave that can go on forever. 𝐸 and 𝐵 are perpendicular to each other, and to the direction of propagation: 𝑣𝐸×𝐵 where the magnitude of 𝑣 is 𝑐.

Electromagnetic Spectrum

Gamma rays--X rays--Light(Ultra-Violet--Visible light--Infra-Red)--MicroWave--Radio(TV--FM--Amateur band--AM--Long Wave)

Accelerated Charges

image Electromagnetic pulse can propagate in space. Information about the charge's wiggle travels outward at the speed of light. The disturbance takes the shape of a sphere. 𝐸radial∼(1/𝑟^3);𝐸transverse∼(1/𝑟), therefore 𝐸transverse≫𝐸radial at large distance.
Transverse pulse propagates at speed of light. Since 𝐸(t), there must be 𝐵. Direction of 𝑣 is therefore given by: 𝐸×𝐵 image Magnitude can be derived from Gauss's law, first shown by Edward Purcell.
Field∼−𝑞𝑎
𝐸radiative=14𝜋𝜀0−𝑞𝑎𝑐2𝑟
The direction of the transverse field is opposite to 𝑞𝑎. The electric field falls off at a rate 1/𝑟.

Why is the Sky Blue

On the Moon, the sky looks black. Gas molecules scatter blue/violet light more than red light. Sky is blue for the same reason, the sunset is red.

Source and Reference

https://www.youtube.com/watch?v=jbHgG1iHoY0&list=PLZ6kagz8q0bvxaUKCe2RRvU_h7wtNNxxi&index=27

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ID: 200200802 Last Updated: 8/2/2020 Revision: 0

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