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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+𝜀0∂𝐸∂𝑡 ⇒∇×𝐵=𝜇0𝜀0∂𝐸∂𝑡.

Wave Equations

Once a wave starts, Both Faraday and Ampere equations keep going. Feed one equation into the other: ∇×∇×𝐵=𝜇0𝜀0∂𝐸∂𝑡
⇒∇×∇×𝐵=𝜇0𝜀0∂∂𝑡∇×𝐸=𝜇0𝜀0∂∂𝑡−∂𝐵∂𝑡=−1𝑐2∂2∂𝑡2𝐵
⇒−∂2∂𝑥2+∂2∂𝑦2+∂2∂𝑧2𝐵=−1𝑐2∂2∂𝑡2𝐵
⇒∂2∂𝑥2+∂2∂𝑦2+∂2∂𝑧2𝐵=1𝑐2∂2∂𝑡2𝐵
and similarly
⇒∂2∂𝑥2+∂2∂𝑦2+∂2∂𝑧2𝐸=1𝑐2∂2∂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
1𝑐²∂²∂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=𝑥 𝑦 𝑧∂∂𝑥 ∂∂𝑦 ∂∂𝑧0 0 𝐸_𝑧=∂∂𝑦𝐸_𝑧𝑥−∂∂𝑥𝐸_𝑧𝑦=−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: 𝑣∥𝐸×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

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. 𝐸_radial∼(1/𝑟^3);𝐸_transverse∼(1/𝑟), therefore 𝐸_transverse≫𝐸_radial 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 Magnitude can be derived from Gauss's law, first shown by Edward Purcell.
Field∼−𝑞𝑎_⟂
𝐸_radiative=14𝜋𝜀₀−𝑞𝑎_⟂𝑐²𝑟
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