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: Conceptualization:
The light intensity in an interference pattern is determined by the phase difference (\(\delta\)) between the interfering waves. Maximum intensity occurs during constructive interference. The objective is to ascertain the condition on \(\delta\) for the provided intensity expression to be maximal.
Step 2: Methodological Framework:
The intensity is quantified by the equation:
\[ I = 4A^2 \cos^2\left(\frac{\delta}{2}\right) \] Maximizing the intensity necessitates finding the maximum value of the \( \cos^2(\delta/2) \) term.
Step 3: Analytical Derivation:
The function \( \cos^2(x) \) attains its peak value of 1. Consequently, the intensity \(I\) is maximized when:
\[ \cos^2\left(\frac{\delta}{2}\right) = 1 \] This condition implies:
\[ \cos\left(\frac{\delta}{2}\right) = \pm 1 \] The cosine function yields \(\pm 1\) 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 \(\delta\):
\[ \delta = 2n\pi \] Thus, the phase difference \(\delta\) must be an even integer multiple of \(\pi\), which is equivalent to an integral multiple of \(2\pi\).
Step 4: Conclusion:
Maximum intensity is achieved when \(\delta\) is an integral multiple of \(2\pi\). This condition is synonymous with constructive interference.