Question

white light containing wavelengths from 410 nm to 750 nm falls on a grating with 7200 slits/cm. part a how wide is the first-order spectrum on a screen 3.30 m away? express your answer to three significant figures and include the appropriate units. δy = value units submit request answer provide feedback

Explanation:

Step1: Find the grating spacing \( d \)

The grating has 7200 slits/cm, so \( d=\frac{1}{7200}\text{ cm}=\frac{1}{7200\times10^{-2}}\text{ m}=\frac{10^{-2}}{7200}\text{ m}\approx1.389\times10^{-6}\text{ m} \)

Step2: Use the grating equation \( d\sin\theta = m\lambda \) for first - order (\( m = 1 \))

For a wavelength \( \lambda \), \( \sin\theta=\frac{\lambda}{d} \). Since \( \theta \) is small (for visible light and typical grating setups), \( \tan\theta\approx\sin\theta\approx\theta \) (in radians). The position \( y \) on the screen is given by \( y = L\tan\theta\approx L\frac{\lambda}{d} \), where \( L = 3.30\text{ m} \) is the distance to the screen.

Step3: Calculate the position for \( \lambda_1 = 410\text{ nm}=410\times10^{-9}\text{ m} \)

\( y_1 = L\frac{\lambda_1}{d}=3.30\times\frac{410\times10^{-9}}{1.389\times10^{-6}} \)
\( y_1 = 3.30\times\frac{410\times10^{-3}}{1.389}\approx3.30\times0.2952\approx0.974\text{ m} \)

Step4: Calculate the position for \( \lambda_2 = 750\text{ nm}=750\times10^{-9}\text{ m} \)

\( y_2 = L\frac{\lambda_2}{d}=3.30\times\frac{750\times10^{-9}}{1.389\times10^{-6}} \)
\( y_2 = 3.30\times\frac{750\times10^{-3}}{1.389}\approx3.30\times0.540\approx1.782\text{ m} \)

Step5: Calculate the width of the spectrum \( \Delta y=y_2 - y_1 \)

\( \Delta y=1.782 - 0.974 = 0.808\text{ m} \) (after re - calculating with more precise steps:
First, \( d=\frac{1}{7200}\text{ cm}=\frac{1}{7200}\times10^{-2}\text{ m}=\frac{1}{720000}\text{ m}\approx1.38889\times10^{-6}\text{ m} \)
\( y_1 = 3.30\times\frac{410\times10^{-9}}{1.38889\times10^{-6}}=3.30\times\frac{410}{1388.89}\approx3.30\times0.2952 = 0.97416\text{ m} \)
\( y_2 = 3.30\times\frac{750\times10^{-9}}{1.38889\times10^{-6}}=3.30\times\frac{750}{1388.89}\approx3.30\times0.5401 = 1.78233\text{ m} \)
\( \Delta y=y_2 - y_1=1.78233 - 0.97416 = 0.80817\text{ m}\approx0.808\text{ m} \)

Answer:

\( \Delta y=\boxed{0.808\text{ m}} \)