91Ó°ÊÓ

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.

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 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.

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.