Step-by-step Solution:

1. Recall the formula for fringe width in Young's Double Slit Experiment (YDSE). The fringe width, denoted by $\beta$, is the distance between two consecutive bright or dark fringes. The formula for fringe width is given by: $\beta = \frac{\lambda D}{d}$ where:

   • $\lambda$ is the wavelength of the light used.
   • $D$ is the distance from the slits to the screen.
   • $d$ is the distance between the two slits.

2. Analyze the experimental conditions. The problem states that the experiment is carried out using green, red, and blue light, one colour at a time. This implies that the experimental setup, i.e., the distance $D$ and the slit separation $d$, remains constant for all three cases.

3. Determine the relationship between fringe width and wavelength. Since $D$ and $d$ are constant, we can see from the formula that the fringe width $\beta$ is directly proportional to the wavelength $\lambda$ of the light used. $\beta \propto \lambda$ This means that the light with the longest wavelength will produce the widest fringes, and the light with the shortest wavelength will produce the narrowest fringes.

4. Compare the wavelengths of red, green, and blue light. In the visible spectrum of light (VIBGYOR: Violet, Indigo, Blue, Green, Yellow, Orange, Red), the wavelength increases from violet to red. Therefore, the wavelengths of red ($\lambda_R$), green ($\lambda_G$), and blue ($\lambda_B$) light are in the following order: $\lambda_R > \lambda_G > \lambda_B$ (Typical approximate wavelengths are: $\lambda_R \approx 650-750$ nm, $\lambda_G \approx 500-570$ nm, $\lambda_B \approx 450-495$ nm).

5. Compare the fringe widths. Since the fringe width $\beta$ is directly proportional to the wavelength $\lambda$, the order of the fringe widths (eta_R, eta_G, eta_B) will be the same as the order of the wavelengths. $\beta_R > \beta_G > \beta_B$

6. Match the result with the given options.

   • A: $\beta$ G > $\beta$ B > $\beta$ R - Incorrect.
   • B: $\beta$ B > $\beta$ G > $\beta$ R - Incorrect.
   • C: $\beta$ R > $\beta$ B > $\beta$ G - Incorrect.
   • D: $\beta$ R > $\beta$ G > $\beta$ B - Correct.

Thus, the correct relationship between the fringe widths is $\beta_R > \beta_G > \beta_B$.