Questions: Blue light (450 nm) and orange light (625 nm) pass through a diffraction grating with d=2.88 x 10^-6 m. What is the angular separation between them for m=1? [?]^° Remember: nano means 10^-9

Solution Steps

The diffraction grating equation is given by: \[ d \sin \theta = m \lambda \] where:

• \( d \) is the distance between adjacent grating lines,
• \( \theta \) is the diffraction angle,
• \( m \) is the order of the diffraction,
• \( \lambda \) is the wavelength of the light.

Given:

• Blue light wavelength, \( \lambda_{\text{blue}} = 450 \, \text{nm} = 450 \times 10^{-9} \, \text{m} \)
• Orange light wavelength, \( \lambda_{\text{orange}} = 625 \, \text{nm} = 625 \times 10^{-9} \, \text{m} \)

Using the diffraction grating equation for \( m = 1 \): \[ \sin \theta_{\text{blue}} = \frac{m \lambda_{\text{blue}}}{d} = \frac{1 \times 450 \times 10^{-9}}{2.88 \times 10^{-6}} \] \[ \sin \theta_{\text{blue}} = 0.1563 \] \[ \theta_{\text{blue}} = \arcsin(0.1563) \approx 8.98^\circ \]

Similarly, for orange light: \[ \sin \theta_{\text{orange}} = \frac{m \lambda_{\text{orange}}}{d} = \frac{1 \times 625 \times 10^{-9}}{2.88 \times 10^{-6}} \] \[ \sin \theta_{\text{orange}} = 0.2170 \] \[ \theta_{\text{orange}} = \arcsin(0.2170) \approx 12.55^\circ \]

The angular separation between the blue and orange light is: \[ \Delta \theta = \theta_{\text{orange}} - \theta_{\text{blue}} \] \[ \Delta \theta = 12.55^\circ - 8.98^\circ \] \[ \Delta \theta = 3.57^\circ \]

Final Answer