Maxwell's Equations & Wave Equation: An Overview

[may de vera]

Let respectively b = (b₁, b₂, b₃) and e = (e₁, e₂, e₃) denote the magnetic
and electric field in some medium. They are governed by Maxwell’s equations which look as follows:
(0.1) \partial_te = curl b
(0.2) \partial_tb = − curl e
(0.3) div e = 0
(0.4) div b = 0.
Show that each b_i and each e_i satisfies the wave equation
(\partial_t² - \Delta) \varphi=0

 

[Dick]

Try it. It's not that hard. It might help to know curl(curl(A))=grad(div(A))-laplacian(A).

 

[may de vera]

taking curl of both side:
curl (dt e) = grad (div e) - laplacian (e)
curl (dt e) = grad (0) - laplacian (e)

i don't know what I'm doing =(

 

[Dick]

dt e=curl b. Taking curl of both sides. curl(dt e)=curl(curl(b))=grad(div B)-laplacian(B)=grad(0)-laplacian(b)=-laplacian(b). Now curl(dt e)=dt(curl e). Continue...

 

[may de vera]

curl (dt e) = -laplacian (b)
dt (curl e) = -laplacian (b)
-dt b = -laplacian (b)
dt b = laplacian (b)
0 = laplacian (b)

and then taking curl of 0.2:
dtb = -curl e
curl (dt b) = curl (-curl e)
dt (curl b) = -curl (curl e)
dt (e) = -grad( div e) - laplacian (e)
= -grad (0) - laplacian (e)
dt (e) = - laplacian (e)
0 = laplacian e

how does that satisfy the wave equation? is it because that they both equal to zero?

 

[Dick]

You aren't being very careful with signs and you are dropping a dt. dt(dt(e))=\partial^2_t E. Why are you turning dt(e) and dt(b) into zero? You want to get dt(dt(b))=laplacian(b) and the same for e.

 

[may de vera]

i finally got down to: dt(dt(b)) = laplacian (b) and dt(dt(e))= -laplacian e

i still don't understand how that applies to the wave equation.

 

[Dick]

i finally got down to: dt(dt(b)) = laplacian (b) and dt(dt(e))= -laplacian e

i still don't understand how that applies to the wave equation.

You've STILL got a sign wrong in the e part. What you've quoted in the problem as the wave equation is what we've been writing as dt(dt(phi))-laplacian(phi)=0. Look up those symbols.

 

[may de vera]

i found out where i made a mistake with that negative sign. i got it now. thank you so so muchhhhh! happy thanksgiving!