Question

Intensity of two sources is same and path difference at point \(A\) and \(B\) are \( \frac{\lambda}{6} \) and \( \frac{\lambda}{3} \) respectively. Ratio of intensity at \(A\) and \(B\) will be.

• A. \(3\)
• B. \(4\)
• C. \( \frac{1}{3} \)
• D. \( \frac{1}{4} \)

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
When two coherent waves of equal intensity interfere, the resultant intensity depends entirely on the phase difference between them at the point of observation. Phase difference is directly proportional to path difference.
Step 2: Key Formula or Approach:
Phase difference: \(\phi = \frac{2\pi}{\lambda} \Delta x\)
Resultant intensity for equal sources: \(I = 4I_0 \cos^2\left(\frac{\phi}{2}\right)\)
Step 3: Detailed Explanation:
At point A: Path difference \(\Delta x_A = \frac{\lambda}{6}\). \[ \phi_A = \frac{2\pi}{\lambda} \left(\frac{\lambda}{6}\right) = \frac{\pi}{3} \, \text{rad} \] Intensity at A: \[ I_A = 4I_0 \cos^2\left(\frac{\pi/3}{2}\right) = 4I_0 \cos^2\left(\frac{\pi}{6}\right) = 4I_0 \left(\frac{\sqrt{3}}{2}\right)^2 = 4I_0 \times \frac{3}{4} = 3I_0 \]
At point B: Path difference \(\Delta x_B = \frac{\lambda}{3}\). \[ \phi_B = \frac{2\pi}{\lambda} \left(\frac{\lambda}{3}\right) = \frac{2\pi}{3} \, \text{rad} \] Intensity at B: \[ I_B = 4I_0 \cos^2\left(\frac{2\pi/3}{2}\right) = 4I_0 \cos^2\left(\frac{\pi}{3}\right) = 4I_0 \left(\frac{1}{2}\right)^2 = 4I_0 \times \frac{1}{4} = I_0 \]
Ratio of intensities: \[ \frac{I_A}{I_B} = \frac{3I_0}{I_0} = 3 \] Step 4: Final Answer:
The ratio of intensity at A and B is \(3\).