Why do electromagnetic waves travel at the speed of light?

Question

I want to prove the fact that EM waves travel at a speed of light.

I would start with this equation:-

$$c= \lambda\nu$$

I am told that this fact proves that the all EM waves travel at the speed of light. I don't see why multiplying the wavelength times frequency give you the speed. Let me take an example and find the speed of $\gamma$ rays.

I would not like to use the formula above because it looks counterintuitive to me.

Instead I would use:-

$$\vec{v}=\frac{\vec{s}}{t}$$

Here I would take a few assumptions.

$$\vec{s}=\lambda$$

But I am not able to figure what should be my $t$. But by using the above formula and fact that the speed will be equal to speed of light I can say that:-

$$t=\lambda c$$

But the problem is that I do not want to use the above formula or fact.

So can anyone prove that:-

$c=\lambda\nu$

And how can we find the speed of light using the formula for average speed as I wrote above without using the $c=\lambda\nu$ and EM waves travel at speed of light.

Just one more thing how can we show that EM waves travel at the speed of light by using the fact $c=\lambda\nu$.

Answer 1

In short, EM waves travel at the speed of light because $c\equiv\frac{1}{\sqrt{\mu_0\varepsilon_0}}$. This quantity comes from the wave equation found by combining Faraday's Law of Induction and Ampere's Circuital Law. If you would like to learn more, you can read this Wikipedia page.

However, if you are wondering why in a different sense, then note that light and EM waves are the same thing, so by definition, we can call the speed of propagation of EM waves as the speed of light.

Answer 2

$c = \lambda f$ is a good way to measure the speed of light and find out how fast it goes. But the question of why light must travel at that speed, or at any given speed, is a deeper question.

You are on the right track with $c = v = s/t$ and $s = \lambda$.

Light is a wave, something light waves in water. Water waves have crests and troughs. EM waves have E fields with an upward max and downward max. In both cases, they move forward.

If you stay still and watch crests go by, you can count the number per second. That is the frequency, $f = \nu/2 \pi$. If you see $f$ crests pass by in $1$ sec, the time for the wave to advance from one crest to the next (1 wavelength, or $\lambda$) is $1/f$.

So

$$c = s/t = \frac{\lambda}{1/f} = \lambda f$$

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This isn't proof that light must travel at a fundamentally important speed. This works for all kinds of waves that travel at many different speeds.

For insight on why the speed of light must always be the same and why this is so important to physics, see Do we know why there is a speed limit in our universe?. As you can see, lots of people have answered from lots of different viewpoints and at various levels of difficulty.