Correct Option: (C) $2 I_0$

Step 1 (Formula / Governing Law & Given Data): [1/3 Mark]

The resultant intensity $I$ of two interfering waves is given by the formula:

$$I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos \phi$$

Given data:

• $I_1 = I_2 = I_0$
• Path difference ($\Delta x$) = $\lambda/4$

Step 2 (Step-by-Step Mathematical Working): [1/3 Mark]

First, calculate the phase difference ($\phi$) using the relation between path difference and phase difference:

$$\phi = \frac{2\pi}{\lambda} \times \Delta x$$ $$\phi = \frac{2\pi}{\lambda} \times \frac{\lambda}{4} = \frac{\pi}{2}$$

Now, substitute the values into the resultant intensity formula:

$$I = I_0 + I_0 + 2\sqrt{I_0 \cdot I_0} \cos\left(\frac{\pi}{2}\right)$$ $$I = 2I_0 + 2I_0 \cos\left(\frac{\pi}{2}\right)$$

Since $\cos(\pi/2) = 0$:

$$I = 2I_0 + 2I_0(0)$$ $$I = 2I_0$$

Step 3 (Final Concluded Answer): [1/3 Mark]

The resultant intensity at the point is $2 I_0$.

💡 EL10 Concept Insight:

Think of the two waves like two people pushing a swing; because they are out of sync by a quarter of a wavelength, they don't cancel each other out (like they would at half a wavelength) nor do they perfectly combine into a super-wave, so you end up with the simple sum of their individual intensities.