Light Waves and Colors Legacy Problem #16 Guided Solution

This problem involves the use of Young's equation in order to calculate a distance between a central bright spot and the fourth bright spot on a screen. The interference pattern is created by green laser light shining through two slits and projected onto the screen 9.85 meters away. We know three numerical values here. We know the distance between the slits and that state is 45 micrometers. So I say d equal 45 micrometers. I also know the wavelength of light lambda is 532 nanometers. And finally this distance of 9.85 meters is L. It's the distance from the slits to the screen where the pattern is projected. So L equal 9.85 meters. The final quantity that I know is the quantity m. It's a whole number or a half number that corresponds to the distance being measured on the screen. The distance I wish to calculate is the distance between the central bright spot and the fourth bright spot. So for bright spots this number is a whole number and from the central to the fourth it's the whole number 4. Now I wish to solve for this distance and it's represented by y in Young's equation. So I'm going to take Young's equation lambda equal yd over ml and rearrange it to solve for y. That would become y equal m times L times wavelength divided by d. Now if I substitute in values of ml, lambda, and d into the equation I'll be able to solve for y. But what's tricky here is it's a mess of units. And so my strategy will be to convert all the quantities to the same units. It doesn't matter what unit it is, but I'm going to pick meters because that might be the easiest one to convert to. So I convert my 45 micrometers for d into meters. And that involves dividing by 10 to the 6th. I get 4.5 times 10 to the negative 5th in meters. And I convert my nanometers to 532 nanometers for wavelength into meters. And that becomes 5.32 times 10 to the negative 7th meters if I divide by 10 to the 9th. Now all quantities are in meters and I can substitute it into the equation y equal ml lambda divided by d. And if I do so I would be going 4 times 9.85 times 5.32 times 10 to the negative 7th divided by the quantity 4.5 times 10 to the negative 5th. And I get 0.4658 meters and I can convert that to centimeters by multiplying by 146.58 centimeters. And I'll round this to three significant digits. Thank you.