1. Use the interference intensity formula for two identical coherent sources:

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

Since the sources are identical, let

$I_1 = I_2 = I_0$

So,

$I = 2I_0(1+\cos\phi)$

Using the identity $1+\cos\phi = 2\cos^2(\phi/2)$,

$I = 4I_0\cos^2\left(\frac{\phi}{2}\right)$

The maximum intensity is

$I_{\max} = 4I_0$

Hence,

$\frac{I}{I_{\max}} = \cos^2\left(\frac{\phi}{2}\right)$

2. Relate phase difference to path difference:

Given path difference

$\Delta x = \frac{7\lambda}{4}$

Phase difference is

$\phi = \frac{2\pi}{\lambda}\Delta x = \frac{2\pi}{\lambda}\cdot \frac{7\lambda}{4} = \frac{7\pi}{2}$

Therefore,

$\frac{\phi}{2} = \frac{7\pi}{4}$

3. Calculate the intensity ratio:

$\frac{I}{I_{\max}} = \cos^2\left(\frac{7\pi}{4}\right)$

Now,

$\cos\left(\frac{7\pi}{4}\right)=\frac{\sqrt{2}}{2}$

So,

$\cos^2\left(\frac{7\pi}{4}\right)=\left(\frac{\sqrt{2}}{2}\right)^2=\frac{1}{2}$

Thus,

$\frac{I}{I_{\max}}=\frac{1}{2}$

4. Check options:

• A: $\frac{1}{2}$ ✅
• B: $\frac{3}{4}$ ❌
• C: $\frac{1}{3}$ ❌
• D: $\frac{1}{4}$ ❌

Therefore, the correct option is A.