Question

Two coherent light waves, each of intensity \( I_0 \), superpose and produce an interference pattern on a screen. Obtain the expression for the resultant intensity at a point where the phase difference between the waves is \( \phi \). Write its maximum and minimum possible values.

Hint:

Interference intensity varies with phase difference \( \phi \) as \( I = 4I_0 \cos^2\left(\frac{\phi}{2}\right) \). Always remember: maxima occur when waves are in phase, and minima when out of phase by \( \pi \).

Answer 1

Let the two interfering waves be of equal amplitude \( A \), and intensity \( I_0 \propto A^2 \). The resultant amplitude at a point with phase difference \( \phi \) is: \[ A_R = 2A \cos\left(\frac{\phi}{2}\right) \Rightarrow I = A_R^2 = 4A^2 \cos^2\left(\frac{\phi}{2}\right) \Rightarrow I = 4I_0 \cos^2\left(\frac{\phi}{2}\right) \] Resultant Intensity: \[ \boxed{I = 4I_0 \cos^2\left(\frac{\phi}{2}\right)} \] Maximum Intensity: When \( \phi = 0, 2\pi, 4\pi, \ldots \), \( \cos\left(\frac{\phi}{2}\right) = 1 \), \[ I_{\max} = 4I_0 \] Minimum Intensity: When \( \phi = \pi, 3\pi, \ldots \), \( \cos\left(\frac{\phi}{2}\right) = 0 \), \[ I_{\min} = 0 \]