Question

module 7b
light of differing wavelengths on a diffraction grating
visible light passes through a diffraction grating that has 900 slits per centimeter, and the interference pattern is observed on a screen that is 2.58 m from the grating.
part a
in the first-order spectrum, maxima for two different wavelengths are separated on the screen by 2.62 mm. what is the difference between these wavelengths?
express your answer in meters.
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Explanation:

Step1: Find the grating spacing \( d \)

The grating has 900 slits per centimeter. So, \( d=\frac{1}{900}\text{ cm/slit}=\frac{1}{900\times100}\text{ m/slit}=\frac{1}{9\times10^{4}}\text{ m/slit}\approx1.111\times 10^{-5}\text{ m/slit} \)

Step2: Recall the diffraction grating formula

For a diffraction grating, the position of the \( m \)-th order maximum is given by \( y = m\frac{\lambda L}{d} \), where \( \lambda \) is the wavelength, \( L \) is the distance from the grating to the screen, and \( m = 1 \) for first - order.

Let the two wavelengths be \( \lambda_1 \) and \( \lambda_2 \), and their corresponding positions on the screen be \( y_1 \) and \( y_2 \). Then \( y_1=m\frac{\lambda_1L}{d} \) and \( y_2=m\frac{\lambda_2L}{d} \) (with \( m = 1 \)).

The difference in their positions \( \Delta y=y_2 - y_1=\frac{L}{d}(\lambda_2-\lambda_1) \)

We can re - arrange this formula to solve for \( \Delta\lambda=\lambda_2-\lambda_1 \): \( \Delta\lambda=\frac{\Delta y\times d}{L} \)

Step3: Substitute the values

We know that \( \Delta y = 2.62\text{ mm}=2.62\times10^{-3}\text{ m} \), \( d=\frac{1}{9\times 10^{4}}\text{ m} \), and \( L = 2.58\text{ m} \)

Substitute these values into the formula for \( \Delta\lambda \):

\( \Delta\lambda=\frac{2.62\times 10^{-3}\text{ m}\times\frac{1}{9\times 10^{4}}\text{ m}}{2.58\text{ m}}=\frac{2.62\times10^{-3}}{9\times10^{4}\times2.58}\text{ m} \)

First, calculate the denominator: \( 9\times10^{4}\times2.58 = 9\times2.58\times10^{4}=23.22\times10^{4}=2.322\times10^{5} \)

Then, \( \Delta\lambda=\frac{2.62\times 10^{-3}}{2.322\times 10^{5}}\text{ m}=\frac{2.62}{2.322\times10^{8}}\text{ m}\approx1.128\times 10^{-8}\text{ m} \)

Answer:

\( 1.13\times 10^{-8}\text{ m} \) (or more precisely \( \approx1.128\times10^{-8}\text{ m} \))