Intensity in double slit interference

[physiks]

I'm a little confused about a small detail when finding the intensity over the screen, as some notes I'm using happen to calculate the same thing twice, with a slight difference the second time.

Each method does the following
Call the electric field at any point E so we have
E=E₀e^i(kx₁-wt)+E₀e^i(kx₂-wt) with x₁,x₂ the paths from each of the two slits to a point on the screen.
Write each path in terms of a common path x starting midway between the slits, so x₁ and x₂ both have a path length difference of δ/2 relative to x. Let x₁ be the shorter path.
Then x₁=x-δ/2 and x₂=x+δ/2, and
E=E₀e^{i(kx-kδ/2-wt)}+E_oe^{i(kx+kδ/2-wt)}.
Factoring out the common terms gives
E=E₀e^i(kx-wt)[2cos(kδ/2)].

This is where this issue is. First it says that the intensity is proportional to E*E whilst later it says it is proportional to E*E/2.

This would give either
4E₀²cos²(kδ/2)->I=4I₀cos²(kδ/2), or
2E₀²cos²(kδ/2)->I=2I₀cos²(kδ/2).
Now the first of these is the expression I always see, so must be right.

I'm a little confused by this. I feel like I might be missing something with regards to time averaging the time varying term as in the E*E/2 method. Can somebody clear this up, thanks :)

 

[ehild]

Factoring out the common terms gives
E=E₀e^i(kx-wt)[2cos(kδ/2)].

This is where this issue is. First it says that the intensity is proportional to E*E whilst later it says it is proportional to E*E/2.

The intensity is proportional to both E² and E²/2 .The intensity of the incident light is I₀= K E₀². The intensity of the diffracted light is I=K [2cos(δ/2)E₀]²=[2cos(δ/2)]² KE₀², that is I=4cos²(δ/2) I₀.

ehild

 

[physiks]

The intensity is proportional to both E² and E²/2 .The intensity of the incident light is I₀= K E₀². The intensity of the diffracted light is I=K [2cos(δ/2)E₀]²=[2cos(δ/2)]² KE₀², that is I=4cos²(δ/2) I₀.

ehild

Why not:

The intensity is proportional to both E² and E²/2 .The intensity of the incident light is I₀= K E₀². The intensity of the diffracted light is I=0.5K [2cos(δ/2)E₀]²=0.5[2cos(δ/2)]² KE₀², that is I=2cos²(δ/2) I₀.

 

[ehild]

Why not:

The intensity is proportional to both E² and E²/2 .The intensity of the incident light is I₀= K E₀². The intensity of the diffracted light is I=0.5K [2cos(δ/2)E₀]²=0.5[2cos(δ/2)]² KE₀², that is I=2cos²(δ/2) I₀.

In that case, I₀=0.5K E₀². Be consistent.

ehild

 

[vela]

This is where this issue is. First it says that the intensity is proportional to E*E whilst later it says it is proportional to E*E/2.

This would give either
4E₀²cos²(kδ/2)->I=4I₀cos²(kδ/2), or
2E₀²cos²(kδ/2)->I=2I₀cos²(kδ/2).
Now the first of these is the expression I always see, so must be right.

In the second line, you replaced $E_0^2$ by $I_0$, but it should be $2(E_0^2/2) = 2 I_0$.