Step 1: Understanding the Concept:
The intensity in an interference pattern varies with the phase difference between the two waves.
The phase difference is derived from the path difference.
We determine the maximum possible intensity ($I_{\text{max}}$) from the first condition and use it to find the intensity for the second condition.
Step 2: Key Formula or Approach:
Relation between Phase difference $\phi$ and Path difference $\Delta x$: $\phi = \frac{2\pi}{\lambda} \Delta x$.
Intensity formula: $I = I_{\text{max}} \cos^2\left(\frac{\phi}{2}\right)$.
Step 3: Detailed Explanation:
Case 1: Path difference $\Delta x_1 = \frac{\lambda}{4}$.
Calculate phase difference $\phi_1$:
\[ \phi_1 = \frac{2\pi}{\lambda} \left(\frac{\lambda}{4}\right) = \frac{\pi}{2} \text{ radians} \]
The intensity $I_1 = K/2$. Using the intensity formula:
\[ I_1 = I_{\text{max}} \cos^2\left(\frac{\pi/2}{2}\right) = I_{\text{max}} \cos^2\left(\frac{\pi}{4}\right) \]
Since $\cos(\pi/4) = 1/\sqrt{2}$, its square is $1/2$.
\[ \frac{K}{2} = I_{\text{max}} \left(\frac{1}{2}\right) \implies I_{\text{max}} = K \]
Case 2: New path difference $\Delta x_2 = \lambda$.
Calculate new phase difference $\phi_2$:
\[ \phi_2 = \frac{2\pi}{\lambda} (\lambda) = 2\pi \text{ radians} \]
Calculate new intensity $I_2$:
\[ I_2 = I_{\text{max}} \cos^2\left(\frac{2\pi}{2}\right) = I_{\text{max}} \cos^2(\pi) \]
Since $\cos(\pi) = -1$, its square is $1$.
\[ I_2 = I_{\text{max}} \cdot 1 = I_{\text{max}} \]
We found $I_{\text{max}} = K$, therefore $I_2 = K$.
Step 4: Final Answer:
The intensity is K.