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Name: Mahmoud
Status: educator
Age: 40s
Location: N/A
Country: N/A
Date: 1/13/2003


Question:
What formula calculates the width of the central fringe in the Young experiment?


Replies:
The central fringe is a maximum when the distance from the two slits to that point is the same so the light from the two slits arrives in phase. The first minimum on either side occurs when the distance from one slit is 1/2 wavelength greater (or less) than the distance from the other slit so the light from the two slits arrives out of phase.

If the screen is far away compared to the distance between the two slits (a very good approximation in general), you can assume the two rays of light are parallel (although they do meet at the screen.) Then the equation for the first minimum is sin (a) = L/2 where a is the angle between light going to the central maximum and light going to the first minimum. L is the wavelength of the length. If you draw a little right triangle where the hypotenuse goes from one slit to the other and one side is perpendicular to the light rays going off at an angle a with respect to the normal to the screen, you should be able to derive this equation. It is also derived in all elementary textbooks treating this physics.

The angle between the rays going to the minima on either side of the central maximum is then twice this angle and so the width of the central fringe is given by W = D tan (2a) where D is the distance from the fringes to the screen.

Best, Dick Plano


Single slit:

The distance from the center of the slit to the first minimum in the diffraction pattern is going to be just one half wavelength longer or shorter than the distance from the slit edges to that same point. In this case, a light ray from the edge would interfere destructively with a light ray from the center; and similarly for other pairs of rays separated by half the slit width.

Double slit:

The distance from the center of one slit to the first minimum is just half a wavelength longer or shorter than the distance from the center of the other slit to the same point. In this case, a light ray from any point in slit 1 would interfere destructively with a light ray from the same point in slit 2.

If you want to derive the actual math, assume the rays are parallel and figure out the angle at which the rays first interfere destructively. Then you can get the width of the central fringe if you know the distance to the screen on which the pattern is cast.

Tim Mooney



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