help with intensity formula for n slits interference

[helloeveryone]

1. The problem statement, all variables and given/known data


somebody help me, i have a hw question: obtaining the formula of calculating the intensity of n slits interfenrece.
2. Relevant equations

the only source is the formula to calculate the intensity of double split interference: I/Io = (E/Eo)^2, and I = 4Io(cos(pie d... (equation 41-7 physics textbook of 5th edition by halliday, and resnick and Krane)

3. The attempt at a solution
I tried everything in my power but it hopeless, i couldn't see the connection between n slits and 2 slits, help me out, thanx

[weejee]

For n slits,

E(t) = E_0 \left[ \cos\omega t + \cos(\omega t +\phi) + ... + \cos\{\omega t + (n-1) \phi\}\right] = E_0 \left[\sum_{k=0}^{n-1} \cos(\omega t + k\phi) \right]

where \phi = \frac {2 \pi d \sin \theta}{\lambda}

Note that

\displaystyle{\sum_{k=0}^{n-1} \cos(\omega t + k\phi) = \Re\left[ \sum_{k=0}^{n-1} e^{i(\omega t + k\phi)}\right] = \Re\left[ e^{i\omega t}\sum_{k=0}^{n-1} e^{i k\phi}\right] = \Re\left[e^{i\omega t} \frac{1-e^{in\phi}}{1-e^{i\phi}}\right] = \Re\left[e^{i(\omega t + \frac{n-1}{2} \phi)} \frac{e^{i\frac{n}{2}\phi}-e^{-i\frac{n}{2}\phi}}{e^{i\frac{1}{2}\phi}-e^{-i\frac{1}{2}\phi}}\right]}

= \displaystyle{\Re \left[e^{i(\omega t + \frac{n-1}{2} \phi)} \frac{\frac{e^{i\frac{n}{2}\phi}-e^{-i\frac{n}{2}\phi}}{2i}}{\frac{e^{i\frac{1}{2}\phi}-e^{-i\frac{1}{2}\phi}}{2i}} \right] = \Re \left[e^{i(\omega t + \frac{n-1}{2} \phi)} \frac{\sin \frac{n}{2}\phi}{\sin \frac{1}{2}\phi}\right] = \frac{\sin \frac{n}{2}\phi}{\sin \frac{1}{2}\phi} \cos \left(\omega t + \frac{n-1}{2} \phi\right)}

I guess you can work out the rest using the above equation?

[helloeveryone]

thanx you so much for helpin me out. if you help me, you mind to help me a bit more: could you tell me how you obtain the formula above (you don't have to write everything in detail, just give me the source that you get the formula, or just give any formula that involve that formula above (is it the formula in the textbook?). And one last thing, what is R value in the note.

[helloeveryone]

oh, nvm, i got it, that was the equation 41-6 in the textbook for n slits.
If i am not mistake, the rest thing i have to do is use equation I(t)/Io = (E(t)/Eo)^2 to find out I(t)
man, your'e genius, lifesaver, i love u man

[weejee]

You flatter me.

R just means the real part of a complex number, and e^ix = cos(x) + i*sin(x) was used in the above equation.

For the answer, I think you need to drop out the cos(omega*t + phase) part to get the amplitude(rather than the full time dependence of the electric field).