Em Waves

Electromagnetic waves in space
See the book “Optics” by Hecht sections 3.2 and 3.3 Speed of light: EM waves travel in a vacuum with speed c = c = 299,792,458 ms-1 exactly They are transverse waves with (vector) E and B field orthogonal to each other, for this section bold indicates vector quanties. When far from the source (many wavelengths) the E and B fields are in phase The vector E x B points in the direction of motion Travelling waves can be written as E = E 0 ⋅ ei ( kx −ωt ) and B = B 0 ⋅ ei ( kx −ωt )
1

µ 0ε 0

Magnitude of B and E In a travelling wave, by using Maxwell’s equations, you can show that the magnitudes of E (E) and B (B) are related (for example Hecht page 42 section 3.2) c= E B

Recall the units: E in V/m and B in Tesla So the magnitude of the B field in the vacuum is tiny compared to the E field Polarisation of an EM wave is defined as the direction of the E field Transport of energy The energy density in the electric field in a vacuum is
1 U E = ε 0 E 2 Jm-3 2

[6.1]

For a magnetic field the energy density is
1 B2 Jm-3 UB = 2 µ0

[6.2]

For an EM field in a vacuum we have B = E/c => E UB = 2µ0 c
1

1

2

=

E2 2µ0c 2

Using c =

µ 0ε 0

implies U B =

ε0E 2
2

=UE

The E and B fields carry the same energy in an EM wave in vacuum. EM energy flow in a vacuum Consider the energy flow in an EM wave in a vacuum: Let the energy density be U Jm-3 Let the energy flow per second across a unit area be S Watts
E

S Watts 1 m2 B

Energy flow for 1m2 in 1 second S = cU Joules (c = speed of light)

2 1 2 1B S = c ε0E + Jm-2s-1 2 2 µ0

Using c =

1

µ 0ε 0

implies S =

EB

µ0

[6.3]

Note: E = |E| and B = |B| In vector terms

S = (E × B) / µ 0 J m-2 s-1 ≡ c 2ε 0 E × B Watts m-2 [6.4]

This means the energy flow is perpendicular to both the E and B fields
The vector S is known as Poynting’s vector

If we write the E and B fields as: E = E0 cos(kx − ωt ) and B = B0 cos(kx − ωt ) we find: S = c 2ε 0 (E × B ) cos 2 (kx − ωt )

Averaged over time the mean