Constructive interference of monochromatic light

[adrian116]

Homework Statement

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.)

Homework 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

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

 

[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.

 

[adrian116]

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?

 

[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.

 

[adrian116]

then , no other equations can be used?
What I need is to derive it myself by that drawing?