# Intensity in double slit interference

Each method does the following

Call the electric field at any point E so we have

E=E_{0}e^{i(kx1-wt)}+E_{0}e^{i(kx2-wt)} with x_{1},x_{2} 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 x_{1} and x_{2} both have a path length difference of δ/2 relative to x. Let x_{1} be the shorter path.

Then x_{1}=x-δ/2 and x_{2}=x+δ/2, and

E=E_{0}e^{i(kx-kδ/2-wt)}+E_{o}e^{i(kx+kδ/2-wt)}.

Factoring out the common terms gives

E=E_{0}e^{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

4E_{0}^{2}cos^{2}(kδ/2)->I=4I_{0}cos^{2}(kδ/2), or

2E_{0}^{2}cos^{2}(kδ/2)->I=2I_{0}cos^{2}(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 ðŸ™‚

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