Question

In Young's double slit experiment, at two points P and Q on screen, waves from slits $S_1$ and $S_2$ have a path difference of 0 and $\lambda/4$ respectively. The ratio of intensities at point P to that at Q will be \dots

• A. 3 : 2
• B. 2 : 1
• C. $\sqrt{2}$ : 1
• D. 4 : 1

Hint:

Memorize these benchmark path differences for YDSE intensities:
$\Delta x = 0 \implies I_{max}$
$\Delta x = \lambda/4 \implies I_{max} / 2$
$\Delta x = \lambda/2 \implies 0$ (Dark fringe)

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
In a YDSE setup, the light intensity at any point on the screen depends entirely on the path difference between the two interfering waves. We must find the ratio of the intensity at the exact center (point P) to the intensity at a specific off-center point (point Q).

Step 2: Key Formula or Approach:
1. Phase difference ($\phi$) is linked to path difference ($\Delta x$) by the formula:
$$\phi = \frac{2\pi}{\lambda} \times \Delta x$$
2. The resultant intensity ($I$) at any point for coherent identical slits is given by:
$$I = I_{max} \cos^2 \left( \frac{\phi}{2} \right) = 4I_0 \cos^2 \left( \frac{\phi}{2} \right)$$

Step 3: Detailed Explanation:
Let's calculate the intensity at point P:
Path difference $\Delta x_P = 0$.
Phase difference $\phi_P = \frac{2\pi}{\lambda} \times 0 = 0$.
Intensity $I_P = I_{max} \cos^2(0 / 2) = I_{max} (1)^2 = I_{max}$.
(This makes sense, as a path difference of 0 creates the central bright maximum).
Let's calculate the intensity at point Q:
Path difference $\Delta x_Q = \lambda / 4$.
Phase difference $\phi_Q = \frac{2\pi}{\lambda} \times \frac{\lambda}{4} = \frac{\pi}{2}$ radians ($90^\circ$).
Intensity $I_Q = I_{max} \cos^2 \left( \frac{\pi/2}{2} \right) = I_{max} \cos^2 \left( \frac{\pi}{4} \right)$.
Since $\cos(\pi/4) = 1/\sqrt{2}$:
$$I_Q = I_{max} \left( \frac{1}{\sqrt{2}} \right)^2 = I_{max} \left( \frac{1}{2} \right) = \frac{I_{max}}{2}$$.
Now, calculate the required ratio $I_P / I_Q$:
$$\text{Ratio} = \frac{I_{max}}{I_{max} / 2} = \frac{1}{1/2} = 2$$
The ratio is exactly 2 : 1.

Step 4: Final Answer:
The ratio of intensities is 2 : 1, matching option (b).