JEE PYQ: Units & Measurements - Question ID b2511ba5a89b (JEE Main 2008)

In the experiment to determine the refractive index of glass using a travelling microscope, the key concept revolves around measuring small distances with high precision. The refractive index ($n$) of a material (like glass) is given by the relation:

Here: - Real thickness is the actual thickness of the glass slab, measured directly. - Apparent thickness is the thickness as seen through the glass due to refraction, which appears less than the real thickness.

To measure these thicknesses accurately, the experiment requires precise determination of the vertical shift in the microscope's focus when observing the top and bottom surfaces of the glass slab. This shift is typically on the order of millimeters or less, demanding a measurement tool with high resolution.

The travelling microscope is equipped with a vernier scale, which allows measurements to be taken with a precision much finer than a standard meter scale. The vernier scale provides an additional level of accuracy (often up to 0.01 mm or 0.1 mm) by using a secondary sliding scale that aligns with the main scale, enabling the measurement of small distances that a standard scale cannot resolve.

Step 1: Understanding the Measurement Requirement
The experiment involves measuring the vertical displacement of the microscope when focusing on:

• The top surface of the glass slab (without the slab).
• The bottom surface of the glass slab (through the slab).
• The top surface again after placing the slab (to account for any shift).

These displacements are small (typically a few millimeters) and require precision to avoid significant errors in the refractive index calculation.

Step 2: Evaluating the Measurement Tools
Let’s analyze the options:

• Option A: Vernier scale provided on the microscope
  The vernier scale is designed for high-precision measurements. It consists of a main scale and a sliding vernier scale that allows readings to be taken to a fraction of the smallest division on the main scale (e.g., 0.1 mm or 0.01 mm). This is ideal for measuring small distances like the apparent and real thicknesses of the glass slab.
• Option B: Standard laboratory scale
  A standard laboratory scale (e.g., meter scale) typically has a least count of 1 mm. This is insufficient for measuring small displacements in the microscope experiment, as the error would be too large.
• Option C: Meter scale provided on the microscope
  While some microscopes may have a meter scale, it is usually for coarse adjustments and lacks the precision needed for this experiment. The least count is still too large (e.g., 1 mm).
• Option D: Screw gauge provided on the microscope
  A screw gauge is used for measuring very small lengths (e.g., diameters of wires) with high precision. However, it is not typically used for measuring vertical displacements in a travelling microscope setup. The vernier scale is the standard tool for this purpose.

Step 3: Precision and Error Analysis
The refractive index calculation is sensitive to errors in measuring the real and apparent thicknesses. For example:

where $\Delta t$ is the difference between real and apparent thickness. If $\Delta t$ is small (e.g., 1 mm), an error of 0.1 mm in measuring $\Delta t$ can lead to a significant error in $n$. Thus, a tool with high precision (like the vernier scale) is essential.

Step 4: Conclusion
Given the need for high precision in measuring small distances, the vernier scale provided on the microscope is the correct tool for this experiment. The other options either lack the necessary precision or are not designed for this specific measurement.

Trap 1: Confusing the Screw Gauge with the Vernier Scale
Students often confuse the screw gauge with the vernier scale because both are used for precise measurements. However, the screw gauge is typically used for measuring diameters or thicknesses of small objects (e.g., wires), while the vernier scale on a travelling microscope is used for measuring linear displacements. In this experiment, the vernier scale is the correct choice.

Trap 2: Overlooking the Need for Precision
Some students may assume that a standard meter scale is sufficient because the distances involved seem large. However, the difference between real and apparent thickness is small, and a high-precision tool is required to minimize errors in the refractive index calculation.

Trap 3: Misidentifying the Microscope's Scale
Students might think that the meter scale on the microscope is the same as the vernier scale. However, the meter scale is for coarse measurements, while the vernier scale is for fine adjustments. Always check the least count of the scale being used.

Exam Tip:
When answering questions about measurement tools in experiments, always consider:

• The magnitude of the quantity being measured.
• The required precision (least count) of the tool.
• The specific design and purpose of the tool (e.g., vernier scale for linear displacements, screw gauge for diameters).

In this case, the vernier scale is the only option that provides the necessary precision for measuring the small distances involved in the refractive index experiment.