White light falls on two narrow slits separated by 0.40 mm . The interference pattern is observe

step 1 This question seems to me like it belongs in the last chapter. This is an interference question.

White light falls on two narrow slits separated by 0.40 mm ....

White light falls on two narrow slits separated by $0.40 \mathrm{mm} .$ The interference pattern is observed on a screen 3.0 $\mathrm{m}$ away. (a) What is the separation between the first maxima for red light $(\lambda=700 \mathrm{nm})$ and violet light $(\lambda=400 \mathrm{nm}) ? \quad$ (b) $\mathrm{At}$ what point nearest the central maximum will a maximum for yellow light $(\lambda=600 \mathrm{nm})$ coincide with a maximum for violet light? Identify the order for each maximum.

Key Concepts

Young's Double-Slit Interference

This concept describes how coherent light passing through two narrow, closely spaced slits produces an interference pattern on a distant screen. Constructive interference occurs when the difference in path lengths from the two slits is an integer multiple of the wavelength (given by the equation d sin? = m?), resulting in bright fringes or maxima. This principle is fundamental in understanding how different wavelengths produce distinct fringe patterns in interference experiments.

Fringe Spacing and Position Calculation

Fringe spacing is the distance between adjacent bright fringes in an interference pattern and is calculated using the formula y = (m?L)/d, where y is the distance from the central maximum, m is the order of the fringe, ? is the wavelength, L is the distance to the screen, and d is the slit separation. This calculation is particularly useful in determining the positions of the bright fringes for different wavelengths, revealing shifts in the interference pattern when colored light is used.

Order of Maximum and Wavelength Relationship

The order of a maximum, denoted by m, indicates the number of wavelengths by which the path difference between the two waves is an integer multiple. In a multiwavelength (white-light) interference setup, each color has its own order of maxima due to differences in ?. Analyzing the relationship between the orders of maxima for different wavelengths enables one to find points where fringes of different colors may coincide, which is critical in solving problems that involve overlapping fringes, as seen with yellow and violet light in the given scenario.

Transcript

00:01 This question seems to me like it belongs in the last chapter.

00:04 This is an interference question.

00:07 But the slits are separated by 0 .4 millimeters.

00:26 The distance to the screen is 3 meters.

00:37 And we're to look for the first maxima for red and violet light.

00:43 So, lambda red is 770 times 10 to the negative 9th power meters.

00:53 And lambda v is, it's not 700, it's just 700 meters.

01:12 Well, the sign of theta is lambda m over d, but m is one.

01:36 I'm going to draw a diagram here.

01:40 Probably should have drawn that bigger, but that's going to be theta.

01:49 This is l here, and let's see, what should i call that up there? i'm going to call this x.

02:07 And so the sign of theta is x over l.

02:18 Therefore, x is l lambda over d.

02:27 So, x for red is going to be, putting that in a calculator, l, which is 3, lambda, red, which is 700 times 10 to the negative 9th power meters over d, which is 0 .004.

02:56 So, x for red, is going to be 5 tenths of a millimeter.

03:22 X for violet, my calculator, change the 7 to a 4.

03:33 That's going to be 3 tenths of a millimeter.

03:46 Now, i noticed that i made a mistake.

03:49 The decimal point up here is in the wrong place.

03:51 That's 4 tenths of a millimeter, not 4 millimeters...

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