Calculate the angles at which the first dark band and the next bright band are formedin the Fraunhofer diffraction pattern of a slit 0.3 mm wide (λ = 5890Å).
Question
Calculate the angles at which the first dark band and the next bright band are formedin the Fraunhofer diffraction pattern of a slit 0.3 mm wide (λ = 5890Å).
Solution
To calculate the angles for the first dark band and the next bright band in the Fraunhofer diffraction pattern, we can use the formula for single-slit diffraction:
θ = λ / b
where:
- θ is the angle of the diffraction pattern,
- λ is the wavelength of the light, and
- b is the width of the slit.
First, we need to convert the given values to the same units. The width of the slit is given in mm, so we convert it to meters: 0.3 mm = 0.3 x 10^-3 m. The wavelength is given in Ångstroms, so we convert it to meters: 5890 Å = 5890 x 10^-10 m.
- Calculate the angle for the first dark band (m=1):
θ_dark = λ / b θ_dark = (5890 x 10^-10 m) / (0.3 x 10^-3 m) θ_dark = 1.963 x 10^-3 rad
To convert this to degrees, multiply by 180/π: θ_dark = 1.963 x 10^-3 rad * (180/π) = 0.112 degrees
- Calculate the angle for the next bright band (m=2):
θ_bright = 2λ / b θ_bright = 2 * (5890 x 10^-10 m) / (0.3 x 10^-3 m) θ_bright = 3.927 x 10^-3 rad
To convert this to degrees, multiply by 180/π: θ_bright = 3.927 x 10^-3 rad * (180/π) = 0.225 degrees
So, the first dark band is formed at an angle of approximately 0.112 degrees, and the next bright band is formed at an angle of approximately 0.225 degrees.
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