# Constructive interference of monochromatic light

1. Nov 29, 2006

1. The problem statement, all variables and given/known data
A source S of monochromatic light and a detector D are both located in air a distance h above a horizontal plane sheet of glass, and are separated by a horizontal distance x. Waves reaching D directly from S interfere with waves that reflect off the glass. The distance x is small compared to h so that the reflection is at close to normal incidence.
a). Show that the condition for constructive interference is $\sqrt{x^2+4h^2}-x=(m+\frac{1}{2})\lambda$, and the condition for destructive interference is$\sqrt{x^2+4h^2}-x=m\lambda$. (Hint: Take into account the phase change on reflection.)

2. Relevant equations
1.$d\sin\theta=m\lambda$ for constructive interference
2.$d\sin\theta=(m+\frac{1}{2})\lambda$ for destructive interference
3.$\phi=\frac{2\pi}{\lambda}(r_2-r_1)$ phase difference related to path difference

3. The attempt at a solution
I have tried to find d as
$d=\sqrt{h^2+(\frac{x}{2})^2}$
and the phase difference as
$\phi=\frac{2\pi}{\lambda}(\sqrt{h^2+(\frac{x}{2})^2}-x)$
but i do not know how this related to the equations (b) 1 and (b) 2

Last edited: Nov 29, 2006
2. Nov 29, 2006

### Edgardo

Hi,

have you made a drawing of the problem? Analyze the geometry.

Hint: What does the condition for constructive interference mean?
What's the general formula for it?

What are equations (b)1 and (b)2?

Note: The formulas that you listed in "Relevant equations" are for interference when light is incident on slits. But your problem is not about slits.

3. Nov 29, 2006

sorry , not (b)1 and (b)2,
it should be from relevant equations.
As you said, am i necessary to derive other equations
for this problem?
And then following the method as when light is incident on slits
to derive the equations for this problem?

4. Nov 29, 2006

### Edgardo

The problem asks you to derive the "condition for constructive interference".
How is this condition defined in general? (It has something to do with path difference.)

Then, make a drawing, because then you can calculate the path difference.

You don't need the equations for the slit.

5. Nov 29, 2006