How Does Slit Spacing Affect Light Diffraction Patterns?

[cuti_pie75]

Imagine that you have an opaque screen with three horizontal very narrow parallel slits in it. The second is a center-to-center distance a beneath the first, and the third is a distance 5a/2 beneath the first. (a)Write a complex exponential expression in terms of δ for the amplitude of the electric field at some point P at an elevation θ on a distant screen where δ=ka sinθ. Prove that I(θ) = I(0)/3 + 2I(0)/9 (cos δ + cos 3δ/2 + cos 5δ/2)
Verify that at θ=0, I(θ)=I(0)

I think i got the first part (a) right for the exponential expression.
E=Eo Re[e^i(α-wt) (1+e^-iδ +e^i3δ/2)]

But my problem is that i don't know how to prove this Intensity equation.

So any help would be greatly appreciated...

 

[wais]

To prove the intensity equation, we first need to understand what it represents. Intensity, denoted by I, is the power per unit area of an electromagnetic wave. In this case, it represents the power per unit area of the electric field at point P on the distant screen.

We can use the expression for the electric field, E, that you have derived in part (a) to find the intensity, I. The intensity is given by the square of the magnitude of the electric field, so we have:

I = |E|^2 = E*E

Substituting the expression for E that you have derived, we get:

I = E*E = (Eo Re[e^i(α-wt) (1+e^-iδ +e^i3δ/2)]) * (Eo Re[e^i(α-wt) (1+e^-iδ +e^i3δ/2)])

Expanding the expression and using the fact that |e^ix| = 1, we get:

I = E*E = Eo^2 Re[e^i(α-wt) (1+e^-iδ +e^i3δ/2)] * Re[e^-i(α-wt) (1+e^iδ +e^-i3δ/2)]

Using the trigonometric identity cos(A+B) = cosAcosB - sinAsinB, we can simplify the expression to:

I = E*E = Eo^2 Re[e^i(α-wt) (1+e^-iδ +e^i3δ/2)] * (1 + cos δ + cos 3δ/2)

Now, we need to use the fact that the intensity is proportional to the square of the amplitude of the electric field. This means that I is directly proportional to Eo^2, so we can write:

I = E*E = Eo^2 Re[e^i(α-wt) (1+e^-iδ +e^i3δ/2)] * (1 + cos δ + cos 3δ/2)
= Eo^2 [Re[e^i(α-wt)]^2 (1+e^-iδ +e^i3δ/2)^2] * (1 + cos δ + cos 3δ/2)
= Eo^2 [1