Question

A diffraction grating has $4.00\cdot {10}^3$ lines $/\mathrm{cm}$ and has white light $(400.-700.\mathrm{nm})$ incident on it. What wavelength(s) will be visible at ${45.0}^{\circ }?$

Short answer

Answer: No visible wavelengths will appear at the given angle of 45.0° for the diffraction grating with 4.00 x 10³ lines/cm.

Step-by-step solution

Calculate the grating spacing

To find the grating spacing (d), we need to divide 1 cm by the number of lines per centimeter. In this case, the grating has 4.00 x 10³ lines/cm, so: $d=\frac{1}{4.00\times {10}^3\text{lines/cm}}$ $d=2.50\times {10}^{-4}\mathrm{cm}$

Convert grating spacing to meters

Since we need to calculate wavelengths in nanometers, it is better to convert the grating spacing into meters: $d=2.50\times {10}^{-4}\mathrm{cm}\times \frac{1\mathrm{m}}{100\mathrm{cm}}$ $d=2.50\times {10}^{-6}\mathrm{m}$

Use the diffraction grating formula to find the order

The formula for the diffraction grating is given by: $dsin(\theta )=m\lambda$ Where $d$ is the grating spacing, $\theta$ is the angle of diffraction, $m$ is the order of the diffraction, and $\lambda$ is the wavelength of the light. In our case, $\theta =45.0°$, and we need to find the wavelength, so we can rearrange the formula: $\lambda =\frac{dsin(\theta )}{m}$ $m=\frac{dsin(\theta )}{\lambda }$

Substitute grating spacing and angle to find the order

Now we substitute the grating spacing ($d=2.50\times {10}^{-6}\mathrm{m}$) and the angle ($\theta =45.0°$) into the formula and find the order: $m=\frac{2.50\times {10}^{-6}\mathrm{m}\cdot sin(45.0°)}{\lambda }$

Determine the wavelengths for each order

Using the minimum and maximum values of the white light, we find the ranges of orders for each wavelength: For ${\lambda }_{min}=400\mathrm{nm}$: $m_{min}=\frac{2.50\times {10}^{-6}\mathrm{m}\cdot sin(45.0°)}{400\times {10}^{-9}\mathrm{m}}\approx 2.48$ For ${\lambda }_{max}=700\mathrm{nm}$: $m_{max}=\frac{2.50\times {10}^{-6}\mathrm{m}\cdot sin(45.0°)}{700\times {10}^{-9}\mathrm{m}}\approx 1.45$ Since $m_{min}$ and $m_{max}$ lie between 1 and 2, we can only use the range of wavelengths for the first order $m=1$ to find the visible wavelengths at $45.0°$.

Calculate the visible wavelengths

Now we can use the formula to find the visible wavelengths for the first-order diffraction ($m=1$): $\lambda =\frac{2.50\times {10}^{-6}\mathrm{m}\cdot sin(45.0°)}{1}$ $\lambda \approx 1.77\times {10}^{-6}\mathrm{m}=1770\mathrm{nm}$ Since 1770 nm falls outside the range of white light ($400-700\mathrm{nm}$), there are no visible wavelengths for the first-order diffraction at ${45.0}^{\circ }$.

Key concepts

Grating Spacing

Understanding the concept of grating spacing is crucial when studying diffraction gratings. Grating spacing, denoted by the symbol d, refers to the distance between adjacent slits or lines in a diffraction grating. It is inversely related to the number of lines per unit length. In essence, if you have a grating with a high number of lines per centimeter, the spacing between these lines will be very small, and vice versa.

When approaching a problem, the first step is often to calculate this grating spacing since it plays a significant role in determining the diffraction pattern. In our exercise, the grating spacing is found by taking the reciprocal of the number of lines per centimeter. Mathematically, we express this as:
$$d=\frac{1}{\text{number of lines/cm}}$$
For a grating with 4,000 lines per centimeter, the spacing is 2.50 x 10^-4 cm. Converting to meters is necessary as it provides consistency when using the SI system for subsequent calculations, particularly in the formula where wavelengths are typically expressed in meters or nanometers.

Wavelength Calculation

Wavelength calculation is at the heart of understanding how a diffraction grating interacts with light. Given that different wavelengths of light will diffract at different angles, calculating the wavelength can be essential for predicting the color or intensity of light in various diffraction orders. For this, we use the diffraction grating equation:
$$d\sin (\theta )=m\lambda$$
This formula correlates the grating spacing d, the angle of diffraction θ, the order of diffraction m, and the wavelength of light λ. To find any one of these variables, we rearrange the equation accordingly.

In the given problem, we needed to calculate the resulting wavelength(s) of light at a specific angle (45.0^°). For this, we isolate λ from the equation:
$$\lambda =\frac{d\sin (\theta )}{m}$$
However, in this case, the calculated wavelength falls outside the visible spectrum, indicating that at the angle given, no visible light is observed. This demonstrates how precise measuring and calculations are integral in wavelength determination and also highlights the importance of range—understanding which wavelengths correspond to visible light (typically 400 - 700 nm).

Diffraction Orders

The term 'diffraction orders' refers to the series of spectra produced by the diffraction grating, numbered according to the integer m in the grating equation. These orders represent different instances where constructive interference occurs for different wavelengths. The first order (m=1) is the first series of diffracted light, the second order (m=2) is the next, and so on. Each of these orders will appear at different angles for the same wavelength of light.

In our exercise, we observed that the possible orders for visible light at 45.0^° fell between the first and second order, which means only the first order could be viable for white light's given range (400 - 700 nm). However, after calculating the exact wavelength that would appear at this angle for the first order, we found it to be outside this range. This leads us to conclude that, while the concept of orders is fundamental to understanding diffraction patterns, the visibility of these orders is subject to the restrictions of the visible light spectrum and the particular conditions of the diffraction setup, such as the angle of observation.