To solve for the ratio of the wavelengths of the two monochromatic radiation \(X\) and \(Y\) when their diffraction orders coincide, we can use the diffraction grating equation:
\[ d \sin(\theta) = m \lambda \]
where:
- \(d\) is the grating spacing (distance between adjacent slits),
- \(\theta\) is the angle of diffraction,
- \(m\) is the order of the maximum,
- \(\lambda\) is the wavelength of the radiation.
Given that the second order maximum for radiation \(X\) coincides with the third order maximum for radiation \(Y\), we can denote their wavelengths as \(\lambda_X\) for \(X\) and \(\lambda_Y\) for \(Y\).
For radiation \(X\), in the second order (\(m = 2\)):
\[ d \sin(\theta) = 2 \lambda_X \]
For radiation \(Y\), in the third order (\(m = 3\)):
\[ d \sin(\theta) = 3 \lambda_Y \]
Since both expressions equal the same \(d \sin(\theta)\) for the same angle \(\theta\), we can set them equal to each other:
\[ 2 \lambda_X = 3 \lambda_Y \]
Now, we can find the ratio of the wavelengths:
\[ \frac{\lambda_X}{\lambda_Y} = \frac{3}{2} \]
Thus, the ratio of the wavelengths of \(X\) and \(Y\) is:
\[ \frac{\lambda_X}{\lambda_Y} = \frac{3}{2} \]
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