Video: Slit Width to Slit Separation Ratio for Double-Slit Diffraction

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.

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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 𝑀 and the distance between the slits 𝑑, then in this problem we want to solve for the ratio of 𝑀 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 𝑀 times the sine of the angle off of the line to the central maximum, πœƒ, is equal to the minimum order number π‘š times the wavelength πœ†.

In our case, because this is the first-order minimum, π‘š is one, so 𝑀 sine πœƒ equals πœ†. If we look next at double-slit diffraction maxima 𝑑, where 𝑑 is the distance between the slits times the sine of πœƒ, is equal to the order number π‘š times πœ†. In our case, because this is the fifth-order maxima, π‘š is five and our equation reads 𝑑 sine πœƒ equals five πœ†.

We can rearrange our top equation to solve for 𝑀 it equals πœ† divided by the sine of πœƒ. Then we rearrange our bottom equation to solve for 𝑑: 𝑑 equals five times πœ† divided by the sine of πœƒ. So if we take the ratio of 𝑀 to 𝑑, then both the πœ† terms and the sine of πœƒ terms cancel out and we’re left with one-fifth. Therefore, 𝑀 to 𝑑 is equal 0.200 to one. That’s the ratio of slit width to distance between the slits.

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