Question

If the definition of SI unit 'X' is required to define SI unit 'Y', the dependency is indicated as \(X \rightarrow Y\). A valid order of dependencies, as per present convention, is __________.

• A. \(\text{metre} \rightarrow \text{second} \rightarrow \text{kilogram} \rightarrow \text{kelvin}\)
• B. \(\text{second} \rightarrow \text{metre} \rightarrow \text{kilogram} \rightarrow \text{kelvin}\)
• C. \(\text{metre} \rightarrow \text{second} \rightarrow \text{kelvin} \rightarrow \text{kilogram}\)
• D. \(\text{second} \rightarrow \text{kilogram} \rightarrow \text{kelvin} \rightarrow \text{metre}\)

Hint:

Look at the SI unit of the constant used to fix each base unit: \(\text{Hz}\) for the second, \(\text{m/s}\) for the metre (via \(c\)), \(\text{kg}\,\text{m}^2/\text{s}\) for the kilogram (via \(h\)), and \(\text{kg}\,\text{m}^2/(\text{s}^2\text{K})\) for the kelvin (via \(k_B\)). Whichever base units appear in that constant's dimensions must already be defined first.

Answer 1

The Correct Option is B

Step 1: Recall which SI base units are defined using which physical constants.
Since the 2019 redefinition, each SI base unit is fixed by giving an exact value to one physical constant. The second is fixed by the caesium-133 hyperfine transition frequency, a definition that stands alone and does not need any other SI base unit.
The metre is fixed by the speed of light, \(c\), through \(c = \text{metre}/\text{second}\), so its definition needs the second to already be defined.
The kilogram is fixed by the Planck constant, \(h\), whose SI unit is \(\text{kg}\,\text{m}^2\,\text{s}^{-1}\), so defining the kilogram from \(h\) needs both the metre and the second already defined.
The kelvin is fixed by the Boltzmann constant, \(k_B\), whose SI unit is \(\text{kg}\,\text{m}^2\,\text{s}^{-2}\,\text{K}^{-1}\), so defining the kelvin from \(k_B\) needs the kilogram, the metre and the second already defined.

Step 2: Turn this into a dependency chain.
Second needs nothing, metre needs second, kilogram needs metre and second, kelvin needs kilogram (and metre and second). A valid order in which the definitions can be built up, each one only using units already defined before it, is:
\[ \text{second} \rightarrow \text{metre} \rightarrow \text{kilogram} \rightarrow \text{kelvin} \]

Step 3: Why the other options are wrong.
Option A (metre then second then kilogram then kelvin) puts metre before second, but the metre's definition through \(c\) needs the second to exist first, so this order is invalid. Option C (metre then second then kelvin then kilogram) has the same metre-before-second problem, and also puts kelvin before kilogram even though the kelvin's definition through \(k_B\) needs the kilogram already fixed. Option D (second then kilogram then kelvin then metre) tries to fix the kilogram right after the second, skipping the metre, but Planck's constant \(h\) has units of \(\text{kg}\,\text{m}^2\,\text{s}^{-1}\), so the metre must already be defined before the kilogram can be tied down through \(h\).

Final Answer:
The valid dependency order is second, then metre, then kilogram, then kelvin. \[ \boxed{\text{second} \rightarrow \text{metre} \rightarrow \text{kilogram} \rightarrow \text{kelvin}} \]