What is the least count of vernier callipers? (a) \(10^{-4} \mathrm{~m}\) (b) \(10^{-5} \mathrm{~m}\) (c) \(10^{-2} \mathrm{~m}\) (d) \(10^{-3} \mathrm{~m}\)

Let's first understand the concept of least count. Least count is the smallest reading that can be measured using any measuring instrument. It gives us an idea of the accuracy of the measuring device. In the case of vernier calipers, the least count is the difference between one main scale reading (MSR) and one vernier scale reading (VSR).

We have been given four options for the least count of vernier calipers: (a) \(10^{-4} \mathrm{~m}\) (b) \(10^{-5} \mathrm{~m}\) (c) \(10^{-2} \mathrm{~m}\) (d) \(10^{-3} \mathrm{~m}\)

We will now analyze each option: (a) \(10^{-4} \mathrm{~m}\) or 0.0001 meters: This is a very small measurement and is equivalent to 0.1 millimeters or 100 micrometers. Vernier calipers can measure very small measurements, but this value is not common for least count. (b) \(10^{-5} \mathrm{~m}\) or 0.00001 meters: This is an even smaller measurement and is equivalent to 10 micrometers. This value is too small and is not commonly found in vernier calipers' least count. (c) \(10^{-2} \mathrm{~m}\) or 0.01 meters: This is a larger measurement and is equivalent to 10 millimeters. This value is too large for the least count of vernier calipers. (d) \(10^{-3} \mathrm{~m}\) or 0.001 meters: This is equivalent to 1 millimeter. Most vernier calipers have a least count of 0.001 meters or 1 millimeter. Thus, this option is the correct answer.

The least count of vernier calipers is \(10^{-3} \mathrm{~m}\) or 0.001 meters (1 millimeter), which is option (d).