State with reason, how the linear width of central maximum will be affected if (i) Monochromatic yellow light is replaced with red light, and(ii) Distance between the slit and the screen is increased.

In a diffraction pattern produced by a single slit, the linear width of the central maximum is related to the wavelength of light and the distance between the slit and the screen. The central maximum is the region between the first minima on either side of the central peak.
Formula for the Linear Width of Central Maximum:
The angular position of the first minima in the diffraction pattern for a single slit is given by:
sin⁡θ=λa\sin \theta = \frac{\lambda}{a}
Where:
• λ\lambda is the wavelength of light,
• aa is the width of the slit,
• θ\theta is the angle of the first minima from the central axis.
The linear width of the central maximum, WW, can be expressed as:
W=2Ltan⁡θW = 2L \tan \theta
Where:
• LL is the distance between the slit and the screen.
For small angles θ\theta, tan⁡θ≈sin⁡θ\tan \theta \approx \sin \theta, so:
W=2LλaW = 2L \frac{\lambda}{a}
Effect of Replacing Yellow Light with Red Light:
• Yellow light has a wavelength λyellow\lambda_{\text{yellow}} (around 580 nm), and red light has a longer wavelength λred\lambda_{\text{red}} (around 650 nm).
• Since the linear width WW is directly proportional to the wavelength λ\lambda, if yellow light is replaced by red light, the wavelength increases.
Thus, when we replace yellow light with red light:
Wred>WyellowW_{\text{red}} > W_{\text{yellow}}
This means the linear width of the central maximum will increase when red light is used instead of yellow light because the longer wavelength causes the diffraction fringes to spread out more.
Effect of Increasing the Distance Between the Slit and the Screen:
• The formula for the linear width of the central maximum W=2LλaW = 2L \frac{\lambda}{a} shows that the width WW is directly proportional to the distance LL between the slit and the screen.
• If the distance LL is increased, the linear width WW will also increase proportionally.
Thus, when the distance between the slit and the screen is increased:
Wnew>WoldW_{\text{new}} > W_{\text{old}}
This means the linear width of the central maximum will increase when the distance between the slit and the screen is increased.
Summary:
1. Replacing monochromatic yellow light with red light: The linear width of the central maximum will increase because red light has a longer wavelength than yellow light, and diffraction is more pronounced for longer wavelengths.
2. Increasing the distance between the slit and the screen: The linear width of the central maximum will also increase, as the width is directly proportional to the distance from the slit to the screen.