To solve the problem:

We are working with a soap film of thickness $t = 627 \, \text{nm}$, refractive index $n = 1.49$, and light in the wavelength range $300 \, \text{nm}$ to $700 \, \text{nm}$.

(a) Fully Constructive Interference

Constructive interference in the reflected light occurs when:

$2 n t = m \lambda$

Where:

• $n$ is the refractive index ($1.49$),
• $t$ is the thickness ($627 \, \text{nm}$),
• $m$ is the interference order ($m = 1, 2, 3, \dots$),
• $\lambda$ is the wavelength of light.

Rewriting for $\lambda$:

$\lambda = \frac{2 n t}{m}$

The range for $\lambda$ is $300 \, \text{nm} \leq \lambda \leq 700 \, \text{nm}$. Substituting $2 n t = 2 \cdot 1.49 \cdot 627 = 1868.46 \, \text{nm}$:

$m = \frac{1868.46}{\lambda}$

To find valid $m$, compute $m$ for the extremes of $\lambda$ and round to the nearest integers:

• At $\lambda = 300 \, \text{nm}$: $m_{\text{max}} = \frac{1868.46}{300} \approx 6.23 \rightarrow m = 6$,
• At $\lambda = 700 \, \text{nm}$: $m_{\text{min}} = \frac{1868.46}{700} \approx 2.67 \rightarrow m = 3$.

Thus, $m = 3, 4, 5, 6$, giving 4 wavelengths for fully constructive interference.

(b) Fully Destructive Interference

Destructive interference in the reflected light occurs when:

$2 n t = \left( m + \frac{1}{2} \right) \lambda$

Rewriting for $\lambda$:

$\lambda = \frac{2 n t}{m + \frac{1}{2}}$

Substituting $2 n t = 1868.46 \, \text{nm}$:

$m + \frac{1}{2} = \frac{1868.46}{\lambda}$

Compute $m + \frac{1}{2}$ for the extremes of $\lambda$:

• At $\lambda = 300 \, \text{nm}$: $m_{\text{max}} + \frac{1}{2} = \frac{1868.46}{300} \approx 6.23 \rightarrow m = 5$,
• At $\lambda = 700 \, \text{nm}$: $m_{\text{min}} + \frac{1}{2} = \frac{1868.46}{700} \approx 2.67 \rightarrow m = 2$.

Thus, $m = 2, 3, 4, 5$, giving 4 wavelengths for fully destructive interference.

Final Answer:

(a) 4 wavelengths for fully constructive interference,
(b) 4 wavelengths for fully destructive interference.

Would you like further clarification or step-by-step details? Here are some related questions:

1. How does the thickness of the film affect interference patterns?
2. Can this calculation be extended to non-visible wavelengths?
3. What happens if the film is illuminated at an angle?
4. How does a change in the refractive index $n$ modify results?
5. What is the role of the phase shift in thin-film interference?

Tip: Always verify the interference condition equations with the boundary conditions of the system!