Question

In biprism experiment, the $4^{\text{th}}$ dark band is formed opposite to one of the slits. The wavelength of light used is [d = separation between slits, D = distance between slits and screen]

• A. $\frac{d^2}{14D}$
• B. $\frac{d^2}{4D}$
• C. $\frac{d^2}{7D}$
• D. $\frac{d^2}{3.5D}$

Hint:

Whenever a fringe forms "opposite one of the slits," always set its position equal to $\frac{d}{2}$. For the $n^{\text{th}}$ dark band, this gives $\frac{d}{2} = \frac{(2n-1)\lambda D}{2d}$, which simplifies to the shortcut formula $\lambda = \frac{d^2}{(2n-1)D}$. Plucking in $n=4$ immediately yields $\frac{d^2}{7D}$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
In a Fresnel biprism interference setup, the $4^{\text{th}}$ minimum (dark band) forms directly opposite one of the virtual slits. We need to find the wavelength ($\lambda$) of the light source using the structural dimensions $d$ and $D$.

Step 2: Key Formula or Approach:
1. The line of symmetry runs down the center of the apparatus. Since the total distance separating the two coherent slits is $d$, each slit is located at a distance of exactly $\frac{d}{2}$ from the central line.
2. The linear distance ($x_n$) from the central bright fringe to the $n^{\text{th}}$ dark band is given by the destructive interference condition:
$$x_n = (2n - 1)\frac{\beta}{2} = (2n - 1)\frac{\lambda D}{2d}$$ where $\beta = \frac{\lambda D}{d}$ is the fringe width.

Step 3: Detailed Explanation:
The problem states that the $4^{\text{th}}$ dark band forms directly opposite one of the slits. This means its physical distance from the central axis matches the slit offset position exactly:
$$x_4 = \frac{d}{2}$$ Now, substitute $n = 4$ into the standard position formula for a dark fringe:
$$x_4 = (2(4) - 1)\frac{\lambda D}{2d} = (8 - 1)\frac{\lambda D}{2d} = \frac{7\lambda D}{2d}$$ Equate these two expressions for $x_4$ since they describe the same point on the screen:
$$\frac{d}{2} = \frac{7\lambda D}{2d}$$ We can cancel the factor of 2 from both denominators:
$$d = \frac{7\lambda D}{d}$$ Cross-multiply to isolate the wavelength variable $\lambda$:
$$d^2 = 7\lambda D \implies \lambda = \frac{d^2}{7D}$$ This matches the mathematical expression given in option (C).

Step 4: Final Answer:
The wavelength of light used is $\frac{d^2}{7D}$, which corresponds to option (C).