EM waves in phase and E/B = c question

[cosmogrl]

My textbook (Serway and Jewett, Physics for Scientists and Engineers) says that Emax/Bmax = E/B = c. And that E and B are in phase. My question is, if they are in phase, they both reach zero at the same time. At that point, E/B = 0/0 and not c. I know I am missing something, but not sure what.

E = Emax cos (kx - wt) and B = Bmax cos (kx - wt), so if they are in phase, they both equal zero at the same time (when cos = 0) and max at the same time. My brain is having a hard time understanding what happens when they both equal zero, how does that tell me that E/B = c?

 

[kuruman]

The way to think of it is that $E(x,t) = E_{max}\cos(kx-\omega t)$, $B(x,t) = B_{max}\cos(kx-\omega t)$ and that $E_{max}=cB_{max}$. Clearly when $E(x,t)$ is zero, so is $B(x,t)$. From the third equation you get $E_{max}/B_{max} = c$. You can take the ratio $E(x,t)/B(x,t)$ and verify that it is equal to $c$ at all points and times except where and when the fields vanish.

On edit: Nothing happens when the fields vanish. Think of this, you have N kids in a room and N pieces of candy. You give out one piece of candy to one kid tell the kid to eat it and then leave the room. Repeat with another kid and so on. Through this process, the ratio of candy pieces to kids is always 1, all the way down to the last kid. What happens to the ratio after the last kid eats the last piece of candy and leaves the room?

 

[jtbell]

Yes, that's a better way to think of the relationship between E and B, as E = cB. In the usual derivation which you can see at e.g.

http://farside.ph.utexas.edu/teaching/em/lectures/node48.html

it actually emerges as $B_{max} = E_{max} / c$. (equation 457 on that page, with different notation for the amplitude of the wave)

 

[cosmogrl]

The way to think of it is that $E(x,t) = E_{max}\cos(kx-\omega t)$, $B(x,t) = B_{max}\cos(kx-\omega t)$ and that $E_{max}=cB_{max}$. Clearly when $E(x,t)$ is zero, so is $B(x,t)$. From the third equation you get $E_{max}/B_{max} = c$. You can take the ratio $E(x,t)/B(x,t)$ and verify that it is equal to $c$ at all points and times except where and when the fields vanish.

On edit: Nothing happens when the fields vanish. Think of this, you have N kids in a room and N pieces of candy. You give out one piece of candy to one kid tell the kid to eat it and then leave the room. Repeat with another kid and so on. Through this process, the ratio of candy pieces to kids is always 1, all the way down to the last kid. What happens to the ratio after the last kid eats the last piece of candy and leaves the room?

I like the candy/kid analogy, but is there a way to show mathematically how E/B = c even when E and B are zero? Or at that point, do we have to do the ratio of the amplitudes?

 

[sophiecentaur]

At that point, E/B = 0/0 and not c.

Nah. That's not the way it works. 0/0 is indeterminate and not meaningful operation. The Limit of E/B as E approaches 0 is still c. That's the basic idea of Calculus.