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 𝑤 and the distance between the slits 𝑑, then in this problem we want to solve for the ratio of 𝑤 to 𝑑. Let’s start by drawing a sketch of this diffraction pattern. We’re 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 we’re 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, let’s 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 𝑤 times the sine of the angle off of the line to the central maximum, 𝜃, is equal to the minimum order number 𝑚 times the wavelength 𝜆.
In our case, because this is the first-order minimum, 𝑚 is one, so 𝑤 sine 𝜃 equals 𝜆. If we look next at double-slit diffraction maxima 𝑑, where 𝑑 is the distance between the slits times the sine of 𝜃, is equal to the order number 𝑚 times 𝜆. In our case, because this is the fifth-order maxima, 𝑚 is five and our equation reads 𝑑 sine 𝜃 equals five 𝜆.
We can rearrange our top equation to solve for 𝑤 it equals 𝜆 divided by the sine of 𝜃. Then we rearrange our bottom equation to solve for 𝑑: 𝑑 equals five times 𝜆 divided by the sine of 𝜃. So if we take the ratio of 𝑤 to 𝑑, then both the 𝜆 terms and the sine of 𝜃 terms cancel out and we’re left with one-fifth. Therefore, 𝑤 to 𝑑 is equal 0.200 to one. That’s the ratio of slit width to distance between the slits.