Electromagnetic wave equations

[drdizzard]

The problem states: Verify by substitution that the following equations are solutions to equations 34.8 and 34.9 respectively.

E=E(max)cos[kx-wt]

B=B(max)cos[kx-wt]

Equations 34.8 and 34.9 are provided in the attachment along with the problem itself as stated in the textbook.

I'm not really sure where to begin with this problem. The instructor and the book didn't give much info regarding how to do it.

 

[drdizzard]

I tried to take the first and second partials of E and B with respect to x and t but it just doesn't seem to get me where I think the problem wants me to go, any help in the right direction would be appreciated.

 

[Andy Resnick]

Can you show what happened when you took the derivatives? It's a strightforward problem- simple crank-turning.

 

[drdizzard]

The partial of E with respect to x is equivalent to -kE(max)sin[kx-wt]
The second partial with respect to x is equal to -k(squared)E(max)cos[kx-wt]

Partial E with respect to t is -wE(max)sin[kx-wt]
Second partial of E with respect to t is -w(squared)E(max)cos[kx-wt]

The first and second partials of B with respect to x are the same as E with E(max) replaced by B(max).

The same is true of the first and second partials of B with repect to t.

 

[alphysicist]

Hi drdizzard,

In your derivatives you have k, \omega, \mu_0, and \epsilon_0. What is \omega/k equal to? What about the product \mu_0\epsilon_0 that appears in the wave equation?

 

[drdizzard]

w/k is equal to c (speed of light/electromagnetic waves in vacuum).

the product of mu and epsilon is equivalent to 1/c^2

So to verify you take the second partial of E with respect to x and set it equal to the product of mu, epsilon, and the second partial of E with respect to t?

 

[alphysicist]

w/k is equal to c (speed of light/electromagnetic waves in vacuum).

the product of mu and epsilon is equivalent to 1/c^2

So to verify you take the second partial of E with respect to x and set it equal to the product of mu, epsilon, and the second partial of E with respect to t?

The wave equation says they must be equal; to verify it in this case you set the two sides equal and show that the equality is always true. (For example, if you do a series of algebraic steps and end up with something like 1=1, then that is always true.)

 

[drdizzard]

I did all the work according to what I think I'm supposed to do with it and its in the attachment.

I came out with c=c for both E and B