[UPSC CSE 2008] Interference (Q206, 20M)

Let the two waves with parallel electric fields be given by E_1 = 2 \cos \left( \bar{k}_1 \cdot \bar{r} - \omega t + \frac{\pi}{3} \right) \text{kV/m}, E_2 = 5 \cos \left( \bar{k}_2 \cdot \bar{r} - \omega t + \frac{\pi}{3} \right) \text{kV/m}. Find the intensity of each beam $ I_1, I_2 $ and also the interference term $ I_{12} $ at a point where their path difference is zero. Calculate the visibility $ V \left( = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}} \right) $ for the interference pattern. [$ \epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/\text{Nm}^2 $, $ \mu_0 = 4\pi \times 10^{-7} \text{ N/A}^2 $]