© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Identify the concept
The question involves single-slit diffraction. The distance between the first dark fringes on either side of the central bright fringe corresponds to the width of the central maximum in the diffraction pattern.
Step 2: Write the formula for the width of the central maximum
The width of the central maximum in a single-slit diffraction pattern is given by:
$$
\text{Width of central maximum} = \frac{2 \lambda D}{a}
$$
where
$$\lambda$$ is the wavelength of light,
$$D$$ is the distance from the slit to the screen, and
$$a$$ is the width of the slit.
Step 3: Substitute the given values
From the question:
Wavelength, $$\lambda = 600\,\text{nm} = 600 \times 10^{-9}\,\text{m}$$
Slit width, $$a = 1\,\text{mm} = 1 \times 10^{-3}\,\text{m}$$
Distance to screen, $$D = 2\,\text{m}$$
Substitute these into the formula:
$$
\text{Width of central maximum} = \frac{2 \times (600 \times 10^{-9}) \times 2}{1 \times 10^{-3}}
$$
Step 4: Simplify the expression
Perform the multiplication and division step by step:
$$
= \frac{2 \times 600 \times 10^{-9} \times 2}{10^{-3}}
= \frac{2400 \times 10^{-9}}{10^{-3}}
= 2400 \times 10^{-9 + 3}
= 2400 \times 10^{-6}\,\text{m}
= 2.4 \times 10^{-3}\,\text{m}
$$
Step 5: Convert to a convenient unit
We know
$$
2.4 \times 10^{-3}\,\text{m} = 2.4\,\text{mm}.
$$
Hence, the distance between the first dark fringes on either side of the central bright fringe is
$$2.4\,\text{mm}.$$
Final Answer
The distance between the first dark fringes on either side of the central bright fringe is 2.4 mm.
