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.
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