Question

In a single slit diffraction experiment, the width of the slit is made double the original width. How does this affect the size and intensity of the central diffraction band? Explain.

Answer 1

We need to understand the relation between the width of the slit used in a single slit experiment with the size and the intensity of the diffraction pattern formed as a result of this slit to find the required solution for this problem easily.

Complete step by step solution:
We are given an initial situation in which a slit of slit width ‘d’ produces a single diffraction pattern on a screed which is ‘D’ distance away from the slit. The size of a diffraction pattern in single slit experiment is given as –
$x=\dfrac{\lambda D}{d}$
For, the central pattern, the size will be twice as the size of a regular pattern, so initially the size of the central diffraction pattern is given by –
$x=\dfrac{2\lambda D}{d}$
Let the intensity of the given initial setup be proportional to the area of the slit A as –

& I\propto A \\\ & \Rightarrow I\propto {{d}^{2}} \\\ \end{aligned}$$  Now, let us consider the second case, where the slit width becomes ‘2d’ and all other parameters are kept the same. The size of the central diffraction pattern will be changed due to the double of the slit width as given below – $$\begin{aligned} & x'=\dfrac{2\lambda D}{2d} \\\ & \Rightarrow x'=\dfrac{x}{2} \\\ \end{aligned}$$ We understand that the size of the central pattern becomes half the initial value when the slit width is doubled. Now, let us see what happens with the intensity of the central pattern – $$\begin{aligned} & I'\propto {{(2d)}^{2}} \\\ & \Rightarrow \dfrac{I'}{I}=\dfrac{4{{d}^{2}}}{{{d}^{2}}} \\\ & \Rightarrow I'=4I \\\ \end{aligned}$$  The intensity of the central pattern becomes four times the initial intensity. **So, the size of the pattern becomes half and the intensity increases by four times when the slit width is doubled in a single slit diffraction.** **Note:** The intensity of the pattern formed is dependent on the amount of light allowed to pass through the slit. As the slit width increases, the amount of light that enters the system increases and as a result, the intensity of the diffraction pattern increases.