\begin{tabular}{l} A diffraction grating is 3 cm long and has 3000 slits. White light shines through it and \ creates fringes on a screen 0.7 m away. The first fringe of the color pewter appears \ 3.4 cm from the central white fringe. What wavelength is this pewter light (in \ nanometers, nm) \ Question 14 \ Saffron light has a wavelength of 440 nm. it shines through a single slit 0.4 mm wide. \ If a screen is 2.5 m away, how WIDE is the central fringe (from lower dark fringe to \ higher dark fringe)? \ Q \ \hline pts \ \hline\end{tabular}

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UpStudy AI · Accepted Answer

To determine the wavelength of the pewter light using the given diffraction grating setup, we'll follow these steps:

### Given Data:
- **Grating Length (L):** 3 cm = 0.03 m
- **Number of Slits (N):** 3000
- **Distance to Screen (D):** 0.7 m
- **Distance to First Fringe (y):** 3.4 cm = 0.034 m
- **Order of Fringe (m):** 1 (first fringe)

### Step 1: Calculate the Grating Spacing (d)
The grating spacing $$ d $$ is the distance between adjacent slits.

$$
d = \frac{L}{N} = \frac{0.03 \, \text{m}}{3000} = 1 \times 10^{-5} \, \text{m} = 10 \, \mu\text{m}
$$

### Step 2: Use the Diffraction Grating Equation
The diffraction condition for the m-th order fringe is given by:

$$
d \sin \theta = m \lambda
$$

For small angles, $$ \sin \theta \approx \tan \theta \approx \frac{y}{D} $$. Therefore:

$$
\lambda = \frac{d \cdot y}{m \cdot D}
$$

### Step 3: Plug in the Values

$$
\lambda = \frac{1 \times 10^{-5} \, \text{m} \times 0.034 \, \text{m}}{1 \times 0.7 \, \text{m}} = \frac{3.4 \times 10^{-7} \, \text{m}}{0.7} \approx 4.857 \times 10^{-7} \, \text{m}
$$

### Step 4: Convert to Nanometers

$$
\lambda \approx 4.857 \times 10^{-7} \, \text{m} = 485.7 \, \text{nm}
$$

### **Final Answer:**
The wavelength of the pewter light is approximately **486  nm**.
The wavelength of the pewter light is approximately 486 nm.

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