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: Understand the overlap condition.
In a biprism (two-source interference) setup, the 6th bright band of wavelength $\lambda_1$ lands exactly where the 7th dark band of wavelength $\lambda_2$ lands. We want $\lambda_1 : \lambda_2$.
Step 2: Position of a bright band.
The $n$th bright band sits at $y_{n,B} = n\,\lambda_1\,\dfrac{D}{d}$. For the 6th bright band, $n = 6$.
Step 3: Position of a dark band.
The $m$th dark band sits at $y_{m,D} = \left(m - \tfrac{1}{2}\right)\lambda_2\,\dfrac{D}{d}$. For the 7th dark band, $m = 7$, giving the factor $6.5$.
Step 4: Equate the two positions.
Since they coincide, $6\,\lambda_1\,\dfrac{D}{d} = 6.5\,\lambda_2\,\dfrac{D}{d}$.
Step 5: Cancel the common geometry.
The setup factor $\dfrac{D}{d}$ is unchanged, so it cancels: $6\,\lambda_1 = 6.5\,\lambda_2$, which gives $\dfrac{\lambda_1}{\lambda_2} = \dfrac{6.5}{6}$.
Step 6: Clear the decimal.
Multiply top and bottom by 2: $\dfrac{\lambda_1}{\lambda_2} = \dfrac{13}{12}$.
\[ \boxed{\lambda_1 : \lambda_2 = 13 : 12\ \text{(option 2)}} \]