Each component of the electric field and magnetic field satisfies the wave equat

Question

Each component of the electric field and magnetic field satisfies the wave equation.

(a) Calling any one of these components , one has in one spatial dimension
2/x2 (1/c2) 2/t2 = 0

Verify that eikx-iwt is a solution to this equation when k and obey a certain condition. What is this condition? How do k and relate to frequency and wavelength?

(b) Now consider the three-dimensional wave equation
2 (1/c2)2/t2 = 0, where 2 =2/x2+2/y2+2/z2.

Try a solution of the form eik*r-iwt and derive the relation between , kx, ky, and kz.

Explanation / Answer

Substituting the given solution,eikx-iwt in the equation

we get,

(ik)^2 -(iw)^2/c2 =0

-k^2+w^2/c^2 =0

k = sqrt(w/c) is the condition

k = 2*pi/wavelength

w = 2*pi*frequency

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