Question

Two identical coherent sources placed on a diameter of a circle of radius \(R\) at separation \(x\) \((x \ll R)\) symmetrically about the centre of the circle. The sources emit identical wavelength \(\lambda\) each. The number of points on the circle with maximum intensity is \((x = 5\lambda)\)

• A. 20
• B. 22
• C. 24
• D. 26

Hint:

Count carefully: \(n = 0\) and \(n = \pm5\) each give 2 points (on the diameter axis), while \(|n| = 1,2,3,4\) each give 4 points (symmetric in all four quadrants).

Answer 1

The Correct Option is A

Step 1: Underlying Concept:
Path difference at point \(P\) on the circle: \(\Delta x = x\cos\theta\). For maxima: \(\Delta x = n\lambda\). Step 2: Explanation:
\(\cos\theta = n\lambda/x = n/5\). Condition: \(|\cos\theta| \leq 1\). So, \(n = 0, \pm1, \pm2, \pm3, \pm4, \pm5\). \(n = 5\): \(\theta = 0^\circ, 180^\circ\) → 2 points. \(n = -5\): same 2 points. For \(|n| = 1,2,3,4\): each gives 4 points → total = \(4 \times 4 = 16\). \(n = 0\): gives 2 points. Total points: \(2 + 16 + 2 = 20\). Step 3: Conclusion:
Number of maxima on the circle \(= 20\).