A grating with 250 grooves / mm is used with an incandescent light source. Assume the visible sp

A grating with 250 grooves / mm is used with an incandescent...

A grating with 250 grooves $/ \mathrm{mm}$ is used with an incandescent light source. Assume the visible spectrum to range in wavelength from $400 \mathrm{nm}$ to $700 \mathrm{nm}$. In how many orders can one see (a) the entire visible spectrum and (b) the short-wavelength region of the visible spectrum?

Key Concepts

Diffraction Grating

A diffraction grating is an optical component with a regular pattern of lines or grooves that splits and diffracts light into several beams traveling in different directions. The directions of these diffracted beams depend on the spacing between the grooves and the wavelength of the incident light. This principle is essential for analyzing the spectral components of light, especially when working with sources that emit a continuous range of wavelengths, such as an incandescent light source.

Grating Equation

The grating equation, m? = d sin?, relates the wavelength (?) of the light, the grating spacing (d), the diffraction order (m), and the angle of diffraction (?). This equation governs the angles at which constructive interference (maxima) occurs for different orders, making it a key tool for predicting which orders can appear for specific wavelengths. The condition that sin? must be less than or equal to 1 sets an upper limit on the possible orders that can be observed for a given wavelength.

Diffraction Order

The diffraction order, denoted by m, represents the multiple of the wavelength path difference that results in constructive interference. Each order corresponds to a distinct set of diffracted beams. The observable orders are limited by the condition derived from the grating equation (specifically, that the sine of the diffraction angle must be less than or equal to one), which determines the maximum order that can exist for a given wavelength range of light.

Wavelength Range and Maximum Order

When considering a range of wavelengths, such as the visible spectrum, the maximum observable order is determined by the smallest wavelength in the range because shorter wavelengths can 'fit' more orders before exceeding the limit of sin? ? 1. For different portions of the spectrum (e.g., the entire visible range versus just the short-wavelength region), this maximum order varies. This concept allows for the determination of how many orders will display all of the desired wavelengths.

Transcript

00:01 In this problem, 250 grooves per millimeter, which is equal to 2 .5 times 35 groups per millimeter.

00:08 Grading is used to observe the diffraction pattern of incandescent lamp with the visual spectrum range that ranges from 400 nanometer, which i am leveling as lambda l to 700 nanometer, which i am labeling as lambda h.

00:25 Now in part a, to find the total number of entire visible spectrum, note from the d -sign theta equals m -lamda, we can actually use m equals d -sign theta over lambda, because we're trying to find total number of entire visible spectrum.

00:57 All right.

00:59 So from here, you can see that increasing the wavelength means decreasing the number of orders of maximum.

01:08 Right? so if you increase the web length here, that's going to decrease the order of number of maximum here.

01:14 So in other words, the total number of orders of the entire visible spectrum is going to be equal to total number of orders of maximum corresponding to long wavelength limit.

01:24 So that means m max is going to be equal to d sine theta max over lambda h long web length limit.

01:46 D sine theta max, theta max is always going to be 90 degrees.

01:50 So i'm going to write down sign 90 degrees over lambda h.

01:59 So we have the value of d and lambda h.

02:02 D was, actually we don't have the value of d, we have to convert n into d.

02:09 So let's do that first...

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