Question

In biprism experiment, $6^{\text{th}}$ bright band with wavelength '$\lambda_1$' coincides with $7^{\text{th}}$ dark band with wavelength '$\lambda_2$', then the ratio $\lambda_1 : \lambda_2$ is (other setting remains the same)

• A. $7 : 6$
• B. $13 : 12$
• C. $12 : 13$
• D. $6 : 7$

Hint:

To handle "bright coincides with dark" questions quickly, replace the bright order directly with its number ($n$) and the dark order with its half-integer value ($m - 0.5$). Then, simply use the inverse relationship equation: $n\lambda_1 = (m - 0.5)\lambda_2$.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
In a Fresnel biprism experiment, the fringe patterns of two different wavelengths are observed. We are told that the position of the $6^{\text{th}}$ bright fringe of the first wavelength matches the position of the $7^{\text{th}}$ dark fringe of the second wavelength. We need to determine the ratio of their wavelengths.

Step 2: Key Formula or Approach:
The position of the $n^{\text{th}}$ bright fringe from the central maximum is given by:
$$y_{n\text{B}} = n\lambda_1\frac{D}{d}$$ The position of the $m^{\text{th}}$ dark fringe from the central maximum is given by:
$$y_{m\text{D}} = (m - 0.5)\lambda_2\frac{D}{d}$$ Since the two bands coincide at the same spatial position on the screen, their respective position coordinates are equal:
$$y_{6\text{B}} = y_{7\text{D}}$$

Step 3: Detailed Explanation:
Set up the equality for the $6^{\text{th}}$ bright band ($n = 6$) and the $7^{\text{th}}$ dark band ($m = 7$):
$$6\lambda_1\frac{D}{d} = (7 - 0.5)\lambda_2\frac{D}{d}$$ Since the geometric setup parameters ($D$ and $d$) remain unchanged for both measurements, the $\frac{D}{d}$ term cancels out from both sides of the expression:
$$6\lambda_1 = 6.5\lambda_2$$ To find the required ratio $\frac{\lambda_1}{\lambda_2}$, rearrange the terms:
$$\frac{\lambda_1}{\lambda_2} = \frac{6.5}{6}$$ Multiply the numerator and denominator by 2 to convert the decimal into a clean integer fraction:
$$\frac{\lambda_1}{\lambda_2} = \frac{13}{12}$$ This simplifies to a definitive ratio of $13 : 12$.

Step 4: Final Answer:
The ratio $\lambda_1 : \lambda_2$ is $13 : 12$, which corresponds to option (B).