Electromagnetic Waves in the Vacuum

Question 1

How do we manipulate Maxwells equations for a vacuum?

Answer 1

Set ρ to 0 and J to zero, as there is no density or flux in a vacuum.

Question 2

How can we combine the 3rd and fourth maxwell equations in a vacuum?

Answer 2

Take the curl of both sides of the ∇XE = -dB/dt equation, and find that it equals -d/dt∇XB, so can sub in the fourth equation here.

Question 3

Which two equations do we find for E and B?

Answer 3

∇^2 E = μ0ε0d^2E/dt^2, ∇^2 B = μ0ε0d^2B/dt^2

Question 4

What is the equation for the speed of light c in terms of μ0 and ε0?

Answer 4

c = 1/sqrt(μ0*ε0)

Question 5

If we consider E and B-fields of the form E = E0exp(i(kr-ωt)) and B = B0exp(i(kr-ωt)), where k = 2π/λ, what is the phase of the waves? What is the k.r part?

Answer 5

Ф = k.r-ωt. The k.r part = const, which is the equation of a plane, hence these are “Plane Waves”.

Question 6

What does k.r equal?

Answer 6

k.r = k(x)x+k(y)y+k(z)z

Question 7

What can we replace ∇ with in Maxwells equations in a vacuum? How is this found?

Answer 7

Replace ∇ with ik and replace the time derivatives with -iω and divide the i’s out to get new Maxwell equations. Can do this because time derivative of exp(i(kr-ωt)) is just -iωthe exponential, and same for x derivative with ik.

Question 8

What do we learn from these new Maxwell equations with k and ω?

Answer 8

E and B are perpendicular to the wave vector k, hence EM waves are transverse. Second 2 equations show that E/B = ω/k = c, and E,B and k are all mutually perpendicular.