Question

Two slits are illuminated by a 375 nm light.
The angle between the zeroth-order bright
band at the center of the screen and the
fourth-order bright band is 15.1◦.
If the screen is 148 cm from the double-slit,
how far apart is this bright band from the central peak?
Answer in units of cm

Answer 1

One bit of information that you are missing is the distance between the two slits, but we can find that with the following equation:\[d=\frac{ m \lambda }{ \sin \left( \theta \right) } \]where d is the distance between slits; m is the order; λ is the wavelength; and is the angle (15.1°). With that you can now find y, the distance between the zeroth order peak and the 4th order peak:\[y=\frac{ m \lambda D }{ d }\]where y is the distance from the zeroth order peak to the mth order peak; λ is the wavelength; D is the distance between the plane of the slits and the screen on which the interference pattern is projected; and d is the distance between the two slits.

Answer 2

I got 38.5547 cm but it is wrong

Answer 3

Try this equation: \[\tan \left( \theta \right)=\frac{ y }{ D }\]The equations I used earlier make use of an approximation that is accurate for small angles:\[\tan \left( \theta \right)\approx \sin \left( \theta \right)\]Using the tangent equation above, y=39.93

Which form to use is dependent on a number of things, including the accuracy needed. Quite often in optics, we use such approximations. In fact, we frequently use the following approximation, called the Small Angle Approximation:\[\tan \left( \theta \right)\approx \sin \left( \theta \right)\approx \theta \]Note that in the small angle approximation, that last term in the equality, θ, is in radians. I've attached a plot showing you how closely the values from those three different calculations match. Note that in the plot, the plot for θ vs. angle is done in radians but is shown in degrees.

The small angle approximation is also used to simplify equations that would be otherwise difficult to use all throughout physics, from equations motion to quantum physics.

Answer 4

I didn't label the plot. The horizontal axis is in degrees while the vertical axis is unitless except for θ. For θ, the vertical axis is in radians.