Electromagnetic Theory: Do these Fields Solve the maxwell equations?

[Raioneru]

1. Consider the fields:

$$\vec{E}$$ = E₀ * cos(kx-wt)$$\vec{e}$$₁

$$\vec{B}$$ = B₀ * cos(kx-wt)$$\vec{e}$$₁

Do these fields solve the maxwell equations? if so, what do they describe?



2. Homework Equations


$$\vec{E}$$ = E₀ * cos(kx-wt)$$\vec{e}$$₁

$$\vec{B}$$ = B₀ * cos(kx-wt)$$\vec{e}$$₁

The Attempt at a Solution

if these functions holds for the maxwell equations then,
$$\nabla$$.$$\vec{B}$$=0
$$\nabla$$.$$\vec{E}$$=0

that is

$$\nabla$$.$$\vec{B}$$=-$$\vec{B}$$0.$$\vec{K}$$*Sin(kx-wt)

this equation is equal to zero only if and only if B0.K = 0 that means, they are perpendicular vectors. that is the constraint.

I guess the $$\vec{e}$$ means that the wave propagates only in the x direction? since e1=<1,0,0>

hum, I really can't tell if that's the appropriate answer, so could you help please ?
thanks in advance

so the question what do they describe, I wrote the propagation of the electromagnetic wave in the x-direction as time increases.

 

[lanedance]

try peforming the differntitaion and see what you get

also shouldn;t you have 4 equations to satisfy?

 

[Raioneru]

Yes I have 4 equations to satisfy, I'll try the differentiation and I'll get back at you

 

[lanedance]

however what you said is correct, the cos term represents propagation in the x dir'n,

so as you have said
$$\vec{B}_0 = B_0(1,0,0)$$
$$\vec{E}_0 = E_0(1,0,0)$$

how do these relate to the differentiation and formula you have given?

 

[Raioneru]

the problem is asking for: what do these functions describe?

so I figured since these are wave functions, they must be describing the propagation of either the electric field or the magnetic field. and since vector e1 is "attached" it means, the propagation of the electric wave, and the magnetic wave in the x-direction

 

[lanedance]

you need to check whther they satisfy Maxwell's equations

as the elctric field, magnetic field & propagation direction are all parallel, i doubt this is the case

 

[Raioneru]

are you sure? because I thought that in this case, the function holds if and only if:

vector k . vector E0 is 0, which means the electric field is perpendicular to the propagation

 

[lanedance]

have a look at how you have written the fields in your first post

propagation direction is x, E field only has e_1 component which i assume is in the x dir'n and B field only has e_1 component which i assume is also in the x dir'n