Which of the following is the most precise device for measuring length
(i) a vernier calipers with 20 divisions on the sliding scale (ii) a screw gauge of pitch Imm and 100 divisions on the circular scale (iii) an optical instrument that can measure length to within a wavelength of light? 

Options

A. Vernier calipers
B. Screw guage
C. Optical instrument
D. All the above

Answer

To determine which of the given devices is the most precise for measuring length, we need to calculate the least count of each device. The least count is the smallest measurement that can be accurately read from a measuring instrument. The device with the lowest least count is the most precise.

### Step-by-Step Solution:

1. **Vernier Calipers:**
- The least count (LC) of a vernier caliper can be calculated using the formula:
\[ LC = 1 \text{ standard division} - 1 \text{ vernier division} \]
- Given that the vernier caliper has 20 divisions on the sliding scale, we can assume the standard division is 1 cm.
- The value of one vernier division (VD) can be calculated as:
\[ VD = \frac{1 \text{ cm}}{20} = 0.05 \text{ cm} \]
- Therefore, the least count is:
\[ LC = 1 \text{ cm} - 0.05 \text{ cm} = 0.95 \text{ cm} \]
- To find the least count in centimeters:
\[ LC = \frac{0.05 \text{ cm}}{10} = 0.01 \text{ cm} \]

2. **Screw Gauge:**
- The least count of a screw gauge can be calculated using the formula:
\[ LC = \frac{\text{Pitch}}{\text{Number of divisions}} \]
- Given that the pitch is 1 mm (or \(1 \times 10^{-3}\) cm) and there are 100 divisions:
\[ LC = \frac{1 \times 10^{-3} \text{ cm}}{100} = 1 \times 10^{-5} \text{ cm} = 0.0001 \text{ cm} \]

3. **Optical Instrument:**
- The least count of the optical instrument is given as the wavelength of light. Assuming the wavelength of visible light is approximately \(6000 \text{ Å} = 6000 \times 10^{-10} \text{ m} = 6 \times 10^{-7} \text{ m} = 0.0000006 \text{ m} = 0.00000006 \text{ cm}\).
- Thus, the least count is:
\[ LC \approx 6 \times 10^{-7} \text{ m} = 0.0000006 \text{ cm} \]

4. **Comparison of Least Counts:**
- Vernier Calipers: \(0.01 \text{ cm}\)
- Screw Gauge: \(0.0001 \text{ cm}\)
- Optical Instrument: \(0.0000006 \text{ cm}\)

5. **Conclusion:**
- The optical instrument has the smallest least count, making it the most precise device for measuring length.

### Final Answer:
The most precise device for measuring length is the optical instrument that can measure length to within a wavelength of light.