Electrodynamics/Electromagnetic Radiation

How Maxwell Fixed Ampère's Law

When Maxwell was looking at Ampere's Law, he realised that changing electric fields acted like currents, but this was not present in its current form. So Maxwell added it, and the full version of Ampere's Law became:

$\mathrm{\nabla }\times 𝐁={\mu }_0𝐉+{\mu }_0{ϵ}_0\frac{\partial 𝐄}{\partial t}$

Using The New Law

When we look at this in a vacuum and combine it to Faraday's Law of Induction:

$\mathrm{\nabla }\times 𝐁={\mu }_0{ϵ}_0\frac{\partial 𝐄}{\partial t}$

$\mathrm{\nabla }\times 𝐄=-\frac{\partial 𝐁}{\partial t}$

And then we solve this, we can see that it's a wave travelling with velocity c.

Sketch Proof

$\mathrm{\nabla }\times 𝐄=-\frac{\partial 𝐁}{\partial t}$

$-\int (\mathrm{\nabla }\times 𝐄)dt=𝐁$

Substituting this into Ampere's Law gives:

$-\mathrm{\nabla }\times \int (\mathrm{\nabla }\times 𝐄)dt={\mu }_0{ϵ}_0\frac{\partial 𝐄}{\partial t}$

$\mathrm{\nabla }\times (\mathrm{\nabla }\times 𝐄)=-{\mu }_0{ϵ}_0\frac{{\partial }^2𝐄}{\partial t^2}$

We now use the following relation:

$\mathrm{\nabla }\times (\mathrm{\nabla }\times 𝐀)=\mathrm{\nabla }(\mathrm{\nabla }\cdot 𝐀)-{\mathrm{\nabla }}^2𝐀$

This gives us:

${\mathrm{\nabla }}^2𝐄={\mu }_0{ϵ}_0\frac{{\partial }^2𝐄}{\partial t^2}$

And similarly:

${\mathrm{\nabla }}^2𝐁={\mu }_0{ϵ}_0\frac{{\partial }^2𝐁}{\partial t^2}$

This should look familiar to you, it's a second derivative with respect to location, equal to a negative second derivative with respect to time. It's a sine function (really imaginary exponential, but it's satisfactory to use sine here). That moves at velocity $c=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}$.

And experimental values of these numbers gave results extremely close to the observed value of speed of light.

This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws. --Maxwell 1865