Question

In Michelson Interferometer the intensity is expressed as \( 4A^2\cos^2\frac{\delta}{2} \), where \( \delta = \frac{2\pi}{\lambda}(2d \cos\theta) \), d being the distance between Mirrors M₁ and M₂. The intensity is maximum, when \( \delta \) is

• A. integral multiple of \(2\pi\)
• B. Integral multiple of \(\pi\)
• C. odd multiple of \(\pi\).
• D. odd multiple of \(\frac{\pi}{2}\).

Hint:

For interference, remember the conditions for intensity maxima and minima.
{Maximum Intensity (Constructive Interference):} Phase difference \( \delta = 2n\pi \), Path difference \( \Delta x = n\lambda \).
{Minimum Intensity (Destructive Interference):} Phase difference \( \delta = (2n+1)\pi \), Path difference \( \Delta x = (n+1/2)\lambda \).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The intensity of light in an interference pattern depends on the phase difference (\(\delta\)) between the interfering waves. The intensity is maximum when the waves interfere constructively. We need to find the condition on \(\delta\) for which the given intensity expression is maximum.
Step 2: Key Formula or Approach:
The intensity is given by the expression:
\[ I = 4A^2 \cos^2\left(\frac{\delta}{2}\right) \] To find the maximum intensity, we need to find the maximum value of the term \( \cos^2(\delta/2) \).
Step 3: Detailed Explanation:
The cosine squared function, \( \cos^2(x) \), has a maximum value of 1.
Therefore, the intensity \(I\) is maximum when:
\[ \cos^2\left(\frac{\delta}{2}\right) = 1 \] This implies:
\[ \cos\left(\frac{\delta}{2}\right) = \pm 1 \] The cosine function is equal to \(\pm 1\) when its argument is an integer multiple of \(\pi\).
\[ \frac{\delta}{2} = n\pi, \quad \text{where } n = 0, \pm 1, \pm 2, \ldots \] Solving for the phase difference \(\delta\):
\[ \delta = 2n\pi \] This means that the phase difference \(\delta\) must be an even integer multiple of \(\pi\), or simply an integral multiple of \(2\pi\).
Step 4: Final Answer:
The intensity is maximum when \(\delta\) is an integral multiple of \(2\pi\). This corresponds to the condition for constructive interference.