Problem 23 - University Physics with Modern Physics

Question

When laser light of wavelength 632.8 nm passes through a diffraction grating, the first bright spots occur at \(\pm\)17.8\(^\circ\) from the central maximum. (a) What is the line density (in lines/cm) of this grating? (b) How many additional bright spots are there beyond the first bright spots, and at what angles do they occur?

Step-by-Step Solution

Answer

(a) 4820 lines/cm; (b) Additional bright spots at \( \pm 36.7^{\circ} \) and \( \pm 64.0^{\circ} \).

Step 1: Convert Wavelength to Meters

Given the wavelength of 632.8 nm, we first convert this to meters for consistency with other SI units. Recall that 1 nm = 10^{-9} m. Thus, \( \lambda = 632.8 \times 10^{-9} \text{ m} = 6.328 \times 10^{-7} \text{ m} \).

Step 2: Use Diffraction Grating Equation

The diffraction grating equation is \( d \sin \theta = m\lambda \), where \( d \) is the distance between the lines on the grating, \( \lambda \) is the wavelength, \( \theta \) is the angle, and \( m \) is the order of the maximum. For the first bright spot (i.e., \( m = 1 \)), \( \theta = 17.8^\circ \).

Step 3: Solve for Line Spacing d

Rearranging the grating equation from Step 2 for \( d \) gives \( d = \frac{\lambda}{\sin \theta} \). Substitute \( \lambda = 6.328 \times 10^{-7} \text{ m} \) and \( \theta = 17.8\degree \) to find \( d = \frac{6.328 \times 10^{-7}}{\sin(17.8\degree)} \approx 2.074 \times 10^{-6} \text{ m} \).

Step 4: Calculate Line Density

The line density is the reciprocal of \( d \) (in cm). Conversion gives \( d = 2.074 \times 10^{-6} \text{ m} = 2.074 \times 10^{-4} \text{ cm} \). Therefore, the line density is \( \frac{1}{2.074 \times 10^{-4}} \approx 4820 \text{ lines/cm} \).

Step 5: Identify Conditions for Additional Bright Spots

For additional bright spots, solve \( d \sin \theta = m \lambda \) for higher \( m \). This equation becomes \( 2.074 \times 10^{-6} \sin \theta = m \cdot 6.328 \times 10^{-7} \).

Step 6: Calculate Additional Orders

Calculate for \( m = 2, 3,... \) until sin\( \theta \) exceeds 1. The maximum order \( m \) is found by \( m = \lfloor \frac{2.074 \times 10^{-6}}{6.328 \times 10^{-7}} \rfloor = 3 \).

Step 7: Determine Additional Bright Spots

The first order was at \( \pm 17.8^{\circ} \). For \( m = 2\), \( \theta_2 = \arcsin(2 \cdot \frac{6.328 \times 10^{-7}}{2.074 \times 10^{-6}}) \approx 36.7^{\circ} \). For \( m = 3\), \( \theta_3 = \arcsin(3 \cdot \frac{6.328 \times 10^{-7}}{2.074 \times 10^{-6}}) \approx 64.0^{\circ} \). No higher orders are possible.

Key Concepts

Wavelength Conversion

When dealing with wavelengths in physics, it's important to maintain consistency with units. Most scientific equations use the International System of Units (SI) to ensure precision and simplicity in calculations. In this exercise, the initial wavelength provided is 632.8 nanometers (nm). To convert this to meters, simply use the conversion factor where 1 nm equals 1 x 10^{-9} meters. Therefore, our given wavelength becomes \( \lambda = 632.8 \times 10^{-9} \text{ m} \), simplifying to \( 6.328 \times 10^{-7} \text{ m} \). By doing this conversion, we make it easier to use the value in subsequent calculations involving the diffraction equation.

Diffraction Equation

The diffraction grating equation plays a crucial role in determining the path of light as it passes through a diffraction grating. This equation is given by \( d \sin \theta = m\lambda \), where \( d \) is the distance between lines on the grating, \( \lambda \) is the wavelength of light, \( \theta \) is the angle formed by the diffracted light, and \( m \) is the order of diffraction.

In this exercise, we are interested in the first order of bright spots so we set \( m = 1 \) with an angle \( \theta = 17.8\degree \). Use this equation to find the line spacing \( d \), which helps further in finding the line density.

Line Density

Line density signifies how many lines are present per unit length on the diffraction grating. It is usually expressed in lines per centimeter (lines/cm). Once the line spacing \( d \) is calculated using the diffraction equation, \( d = \frac{\lambda}{\sin \theta} \), rearranging this for line density gives the reciprocal of the line spacing, \( \frac{1}{d} \).

In this scenario, after calculating and converting \( d \) to centimeters, \( d = 2.074 \times 10^{-4} \text{ cm} \), the line density becomes \( \frac{1}{2.074 \times 10^{-4}} \text{ lines/cm} \). This corresponds to approximately 4820 lines/cm, indicating the precision of the grating and its ability to separate different wavelengths.

Order of Bright Spots

Diffraction grating not only influences the angle at which light is diffracted but also determines the number of diffraction orders possible. The order (\( m \)) refers to the number of wavelengths by which paths differ leading to constructive interference. For the first bright spot, this is \( m = 1 \).

The maximum order that can be achieved without surpassing the condition \( \sin \theta \leq 1 \), is found using the diffraction equation. Beyond the first order \( (m = 1) \), additional orders \( m = 2, 3 \) occur at angles \( \theta \) calculated using \( \theta = \arcsin(m \cdot \frac{\lambda}{d}) \). These conditions yield angles of approximately 36.7° for \( m = 2 \) and 64.0° for \( m = 3 \). Therefore, under the given setup, there are no additional bright spots at higher orders since it exceeds the sine condition.