Question

If in Young's double slit experiment, fifth bright fringe is located at a distance of 0.3 mm from the central bright fringe, then the distance of the seventh dark fringe from the central bright fringe is:

• A. 0.51 mm
• B. 0.39 mm
• C. 0.45 mm
• D. 0.48 mm

Hint:

In Young's double-slit problems, it's often efficient to first establish the value of the constant $\frac{\lambda D}{d}$ using the initial given information. This constant represents the fringe width ($\beta$) for bright fringes, as $\beta = \frac{\lambda D}{d}$. Then, use this constant to calculate other required fringe positions. Pay attention to whether the question asks for bright or dark fringes, as their formulas differ.

Answer 1

The Correct Option is B

Step 1: Identify the Formulas for Fringe Positions
In Young's double-slit experiment, the positions of bright and dark fringes are given by:
The distance of the $n^{th}$ bright fringe ($y_n^{\text{bright}}$) from the central bright fringe is: \[ y_n^{\text{bright}} = \frac{n \lambda D}{d} \] The distance of the $m^{th}$ dark fringe ($y_m^{\text{dark}}$) from the central bright fringe is: \[ y_m^{\text{dark}} = \frac{(2m - 1) \lambda D}{2d} \] Where: \begin{itemize} \item $\lambda$ is the wavelength of light. \item $D$ is the distance between the slits and the screen. \item $d$ is the distance between the two slits. \item $n$ is the order of the bright fringe (e.g., $n=1$ for the first bright fringe, $n=5$ for the fifth bright fringe). \item $m$ is the order of the dark fringe (e.g., $m=1$ for the first dark fringe, $m=7$ for the seventh dark fringe). \end{itemize} Step 2: Use the Given Information for the Fifth Bright Fringe
We are given that the fifth bright fringe ($n=5$) is located at a distance of $0.3 \text{ mm}$ from the central bright fringe.
So, we have: \[ y_5^{\text{bright}} = 0.3 \text{ mm} \] Substitute $n=5$ into the bright fringe formula: \[ 0.3 \text{ mm} = \frac{5 \lambda D}{d} \] From this equation, we can determine the value of the constant term $\frac{\lambda D}{d}$: \[ \frac{\lambda D}{d} = \frac{0.3 \text{ mm}}{5} \] \[ \frac{\lambda D}{d} = 0.06 \text{ mm} \] Step 3: Compute the Distance of the Seventh Dark Fringe
Now, we need to find the distance of the seventh dark fringe. For the seventh dark fringe, $m=7$.
Substitute $m=7$ into the dark fringe formula: \[ y_7^{\text{dark}} = \frac{(2 \times 7 - 1) \lambda D}{2d} \] \[ y_7^{\text{dark}} = \frac{(14 - 1)}{2} \frac{\lambda D}{d} \] \[ y_7^{\text{dark}} = \frac{13}{2} \frac{\lambda D}{d} \] Now, substitute the value of $\frac{\lambda D}{d}$ calculated in Step 2: \[ y_7^{\text{dark}} = \frac{13}{2} \times (0.06 \text{ mm}) \] \[ y_7^{\text{dark}} = 13 \times 0.03 \text{ mm} \] \[ y_7^{\text{dark}} = 0.39 \text{ mm} \] Step 4: Analyze Options
\begin{itemize} \item Option (1): 0.51 mm. Incorrect. \item Option (2): 0.39 mm. Correct, as it matches our calculated distance. \item Option (3): 0.45 mm. Incorrect. \item Option (4): 0.48 mm. Incorrect. \end{itemize}