Question

Calculate the angles at which the first dark band and the next bright band are formed in the Fraunhoffer diffraction pattern of a slit 0.3mm wide(5890 wavelength)
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Answer 1

0.11 degrees and 0.28 degrees

Answer 2

Step-by-step Explanation:

Given: width of the slit (a) = 0.3 mm

the wavelength of the monochromatic light ( $\lambda$ ) = 5890 $A^{o}$

To Find: angle ( $\theta$ ) of formation of the first dark band and next bright band

Solution:

• Calculating the angle of the first dark band

In Fraunhofer diffraction, the angle of the first dark band can be calculated by the following relation;

$a \ sin(\theta_{n} ) = n \lambda$

where a is the slit width and n is the integer number.

To calculate the first dark band, put n=1 and all the given values, we get;

$(0.3 \times 10^{-3} ) \ sin(\theta_{1} ) = 5890 \times 10^{-10}$

$\Rightarrow sin(\theta_{1} ) = \frac{5890 \times 10^{-10}}{0.3 \times 10^{-3}} = 0.00196$

$\Rightarrow \theta_{1} = sin^{-1} (0.00196) = 0.1123^{o}$

• Calculating the angle of the next bright band

And, the angle of the next bright band can be calculated by the following relation;

$a \ sin(\theta_{n} ) = (2n+1) \frac{\lambda}{2}$

where a is the slit width and n is the integer number.

To calculate the next (2nd) bright band, put n=2 and all the given values, we get;

$(0.3 \times 10^{-3} ) \ sin(\theta_{2} ) = (2(2)+1) \frac{5890 \times 10^{-10}}{2}$

$\Rightarrow sin(\theta_{1} ) = (\frac{5}{2}) (\frac{5890 \times 10^{-10}}{0.3 \times 10^{-3}}) = 0.0049$

$\Rightarrow \theta_{2} = sin^{-1} (0.0049) = 0.2807^{o}$

Hence, the first dark band will be at $\theta_{1} = 0.1123^{o}$ and the next bright maxima will occur at $\theta_{2} = 0.2807^{o}$