Why Is E Equals cB
Question
Why does a vacuum electromagnetic wave satisfy E = cB?
Short Answer
For a plane electromagnetic wave in vacuum,
[ \vec k \times \vec E = \omega \vec B ]
and because E is perpendicular to k, the magnitudes satisfy
[ kE = \omega B ]
so
[ E = \frac{\omega}{k} B ]
For an electromagnetic wave in vacuum, the phase velocity is
[ \frac{\omega}{k} = c ]
therefore
[ E = cB ]
More precisely, this is a relation between amplitudes:
[ E_0 = c B_0 ]
Main Idea
The electric field and magnetic field are not independent in a vacuum wave.
They must satisfy Maxwell’s equations simultaneously, and that forces their amplitudes to be related by the wave speed.
In vacuum the wave speed is the speed of light c, so the ratio becomes
[ \frac{E}{B} = c ]
Why The Units Still Make Sense
This does not mean the units of E and B are identical.
Instead, it means their ratio has the value c:
[ \frac{E}{B} = c ]
So the statement is about the physical relation between their magnitudes inside a vacuum electromagnetic wave.
Physical Intuition
- a changing electric field produces a magnetic field
- a changing magnetic field produces an electric field
- in a self-propagating vacuum wave, the sizes of
EandBmust match the propagation speed - because the propagation speed is
c, the relation becomesE = cB
Counterpoints and Gaps
- this relation is specific to electromagnetic waves in vacuum
- in media, the wave speed is generally not
c, so the corresponding relation changes