1. Nov 25, 2008

### 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 bi and each ei satisfies the wave equation
($$\partial$$t2 - $$\Delta$$) $$\varphi$$=0

2. Nov 25, 2008

### Dick

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

3. Nov 26, 2008

### may de vera

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

i dont know what i'm doing =(

4. Nov 26, 2008

### 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...

5. Nov 26, 2008

### 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?

6. Nov 26, 2008

### 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.

7. Nov 26, 2008

### may de vera

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

i still dont understand how that applies to the wave equation.

8. Nov 26, 2008

### Dick

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.

9. Nov 26, 2008

### 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!