Light waves in Interference?

Question

Light with wavelength 442nm passes throught a double slit sustem that has a slit separation d=0.400mm. Determine how far away a screen must be placed so that dark fringes appear directly opposite of both sides.

Please just show the steps, I don't require a numbercal answer.

Thank you in advance.

Answer 1

I think your question has multiple answers since Yminimum is not given.

Often one of the following: the Y minimum, the wavelength or the slit separation is asked but not the distance from the screen to the slit.


Use Equation 9 of Page 5 (of the attached PDF file) to solve for L.

L = (2 d y) / λ

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Note:
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Good Luck,
Rene

Answer 2

This can be done using the Pythagorean Theorem.

First, draw a figure as follows:
1. Draw a vertical line on the left side of a sheet of paper, about 1 inch long. Place a dots at each end of the line (top and bottom) Each dot represents one of the slits.
2. From the lower of the two dots, draw a horizontal line to the right. I suggest making this line 3 or 4 times longer than the distance between the two dots.
3. From the upper dot on the left, draw a diagonal line to the end of the horizontal line you drew in step 2.

You have now drawn a right triangle. The right angle is at the lower left, and the hypotenuse is the diagonal line you drew in Step 3.

Next, let's use what we know(?) about dark fringes and destructive interference, in order to come up with an expression for the sides of the triangle.

Destructive interference occurs when one path is 1/2 of a wavelength shorter or longer than a second path. We can express this as follows:
We can say that the horizontal line has a length of
N * lambda
where N is a number (not necessarily an integer) to be solved for,
and "lambda" is the wavelength, 442 nm.

For the diagonal line, we can say the length is 1/2 of a wavelength longer, or
(N + 1/2) * lambda

Label the horizontal and diagonal lines with these two expressions. Label the line segment between the two dots as "d".

Finally, use the Pythagorean theorem to write an equation relating the sides of this right triangle. Solve this equation for "N".

Your answer is the length of the horizonal line, so multiply N times lambda to get this distance.

Good luck.