The sodium yellow doubled has wavelength 5890 Å and `lamda` Å`(lamda gt gt 5890"Å")` and resolving power of a grating to resolve these lines is 982, then value of `lamda` is

Options

A. `5896Å`
B. `5880Å`
C. `5869Å`
D. `5876Å`

Answer

To solve the problem, we need to find the value of the wavelength \( \lambda \) given the resolving power of a grating and the known wavelength of the sodium yellow doublet.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Wavelength of the sodium yellow doublet, \( \lambda_1 = 5890 \, \text{Å} \)
- Resolving power, \( R = 982 \)
- We need to find \( \lambda \) such that \( \lambda \gg 5890 \, \text{Å} \).

2. **Understanding Resolving Power:**
- The resolving power \( R \) of a grating is given by the formula:
\[ R = \frac{\lambda}{\Delta \lambda} \]
where \( \Delta \lambda \) is the difference in wavelength that can be resolved.

3. **Express \( \Delta \lambda \):**
- Rearranging the formula gives:
\[ \Delta \lambda = \frac{\lambda}{R} \]

4. **Set Up the Equation:**
- Since \( \Delta \lambda \) represents the difference between the two wavelengths, we can express it as:
\[ \Delta \lambda = \lambda - 5890 \]
- Therefore, we can write:
\[ \lambda - 5890 = \frac{\lambda}{982} \]

5. **Multiply Both Sides by 982:**
- This gives:
\[ 982(\lambda - 5890) = \lambda \]
- Simplifying this, we get:
\[ 982\lambda - 5779180 = \lambda \]

6. **Rearranging the Equation:**
- Bringing all terms involving \( \lambda \) to one side:
\[ 982\lambda - \lambda = 5779180 \]
- This simplifies to:
\[ 981\lambda = 5779180 \]

7. **Solve for \( \lambda \):**
- Dividing both sides by 981:
\[ \lambda = \frac{5779180}{981} \approx 5896.0 \, \text{Å} \]

8. **Final Answer:**
- Rounding to the nearest whole number, we find:
\[ \lambda \approx 5896 \, \text{Å} \]

### Conclusion:
The value of \( \lambda \) is approximately \( 5896 \, \text{Å} \).