Video Transcript
A double slit produces a diffraction pattern that is a combination of single- and double-slit interference, where the first minimum of the single-slit pattern falls on the fifth maximum of the double-slit pattern. Find the ratio of the width of the slits to the distance between the slits.
If we call the width of the slits π€ and the distance between the slits π, then in this problem we want to solve for the ratio of π€ to π. Letβs start by drawing a sketch of this diffraction pattern. Weβre told that in this problem we see an overlay of a double-slit diffraction pattern, with a single slit pattern, on top of a single-slit pattern. And weβre told that the first minimum of the single-slit pattern coincides with the fifth maximum of the double- slit pattern.
To find the ratio of the slit width to the distance between the slits, letβs look in turn at the equation for the minimum of the single-slit diffraction pattern and then the maximum of a double-slit pattern. First, looking at the single-slit diffraction minima equation, slit width π€ times the sine of the angle off of the line to the central maximum, π, is equal to the minimum order number π times the wavelength π.
In our case, because this is the first-order minimum, π is one, so π€ sine π equals π. If we look next at double-slit diffraction maxima π, where π is the distance between the slits times the sine of π, is equal to the order number π times π. In our case, because this is the fifth-order maxima, π is five and our equation reads π sine π equals five π.
We can rearrange our top equation to solve for π€ it equals π divided by the sine of π. Then we rearrange our bottom equation to solve for π: π equals five times π divided by the sine of π. So if we take the ratio of π€ to π, then both the π terms and the sine of π terms cancel out and weβre left with one-fifth. Therefore, π€ to π is equal 0.200 to one. Thatβs the ratio of slit width to distance between the slits.
