Question

In a grating with grating constant $d = a + b$, where $a$ is the slit width and $b$ is the separation between the slits, the diffraction pattern has the fourth order missing. The value of $\frac{b}{a}$ is ........... (Specify your answer as an integer.)
 

Hint:

Missing orders occur when interference maxima overlap diffraction minima.

Answer 1

Correct Answer: 3

Step 1: Condition for missing order.
A missing (absent) order occurs when the interference maximum coincides with a single-slit minimum.

Step 2: Write the conditions.
Interference maxima: $d \sin\theta = m\lambda$.
Single-slit minima: $a \sin\theta = n\lambda$.

Step 3: Eliminate $\sin\theta$.
$\frac{m}{n} = \frac{d}{a} = \frac{a+b}{a} = 1 + \frac{b}{a}$.

Step 4: For fourth order missing, $m = 4$ and $n = 1$.
Thus, $4 = 1 + \frac{b}{a}$, giving $\frac{b}{a} = 3$.