Question

The resolving power of a grating:
A. increases with increase in total number of lines ruled on grating.
B. increases with increase in total width of grating.
C. increases with increasing the order of spectrum as in Echelon grating.
D. increases with decreasing the order of spectrum as in Echelon grating.
Choose the CORRECT answer from the options given below:

• A. A, B and D only
• B. A, B and C only
• C. B and D only
• D. A and D only

Hint:

To get high resolution with a grating, you want to use a grating with many total lines (\(N\)) and observe the spectrum in a high order (\(m\)). Both factors directly improve the ability to separate close spectral lines.

Answer 1

The Correct Option is B

Step 1: Recall the formula for the resolving power of a diffraction grating. The resolving power \(R\) is defined as \(R = \frac{\lambda}{\Delta\lambda}\), where \(\Delta\lambda\) is the smallest wavelength difference that can be distinguished at wavelength \(\lambda\). The formula for a grating is: \[ R = mN \] where \(m\) is the spectral order (an integer) and \(N\) is the total number of lines (or rulings) on the grating.
Step 2: Analyze the given statements based on the formula. - A. increases with increase in total number of lines (N): From \(R=mN\), it is clear that if \(N\) increases, \(R\) increases. This statement is correct. - B. increases with increase in total width of grating (W): The total width \(W\) is related to \(N\) and the grating element \(d\) by \(W = Nd\). The grating equation is \(d\sin\theta = m\lambda\). Substituting \(d = W/N\) gives \(\frac{W}{N}\sin\theta = m\lambda \implies mN = \frac{W\sin\theta}{\lambda}\). So, \(R = \frac{W\sin\theta}{\lambda}\). Thus, resolving power is directly proportional to the total width \(W\). This statement is correct. - C. increases with increasing the order of spectrum (m): From \(R=mN\), if \(m\) increases, \(R\) increases. An Echelon grating is specifically designed to work at very high orders (\(m\)), giving it very high resolving power. This statement is correct. - D. increases with decreasing the order of spectrum: This is the opposite of C and is incorrect.
Step 3: Conclude the correct statements. Statements A, B, and C are correct descriptions of the factors affecting the resolving power of a grating.