Light wave E(x,t)=(E_o)exp[i(kx-wt)]

[Liquidxlax]

Homework Statement

Consider the light wave E(x,t) = E_oe^i(kx-wt) propagating through a glass plate of thickness d and refractive index n. Assume the refractive index of the surrounding medium is 1.


|___(n)___|__________________________________|_____>X
x=0 x=d x=L

a) In the absence of the glass plate, show that the field at x=L is related to the field at x=0 by

E(L,t) = E(0,t)e^ia find a

b) With the glass plate inserted, the new field at x=L is related to the incident field at x=0 by

E'(L,t) = E(0,t)e^ia' find a'

(hint: you may want to divide the problem up into two parts. First find E(d,t) in terms of E(0,t), then find E(L,t) in terms of E(d,t).)

The Attempt at a Solution

a) a=kL because wt is the same for both, or so i think



b) I'm not totally sure what to insert here considering my textbook is lacking on this part. What i assume is that since

k=2pi/wavelength and the change in wavelength in a medium is the wavelength divided by the index of refraction. so

nk=n2pi/wavelength

E(d,t)= E_oe^nkd-wt

so for E(0,t) to equal E(d,t). E(0,t) would have to be multiplied by e^ndk


thanks in advance, hope this is understandable

 

[anathema]

Hello,

Your approach to part a) is correct. Since the wave is propagating through the same medium, the phase difference between the two points (x=0 and x=L) will be the same and therefore the phase constant a will be equal to kL.

For part b), you are on the right track but there are a few things that need to be clarified. First, the phase constant a' will not be equal to kL because the wave is now propagating through a different medium (the glass plate). The correct expression for a' will involve the refractive index of the glass plate and the thickness d.

Secondly, in order to find E(d,t) in terms of E(0,t), you will need to use the fact that the wave is traveling through a medium with a different refractive index. This will change the wave vector k and the angular frequency w of the wave. You can use the relation you mentioned, nk=n2pi/wavelength, to find the new values of k and w in terms of the original values.

Once you have found E(d,t) in terms of E(0,t), you can use this result to find E(L,t) in terms of E(d,t). This will involve another phase constant that will be dependent on the refractive index of the glass plate and the thickness d. Combining this result with the expression you found for E(L,t) in part a) will give you the final expression for E'(L,t) in terms of E(0,t).

I hope this helps clarify the steps needed to solve part b). Good luck!