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Difficulty: HardFundamental and Derived Units

The derived SI unit of electrical resistance is the ohm (Ω\Omega). Which of the following correctly expresses the ohm strictly in terms of fundamental SI base units?

  1. kg⋅m2⋅s−3⋅A−2\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-2}Answer
  2. B
    kg⋅m2⋅s−2⋅A−2\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}\cdot\text{A}^{-2}
  3. C
    V⋅A−1\text{V}\cdot\text{A}^{-1}
  4. D
    kg⋅m2⋅s−3⋅A−1\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}

Answer

kg⋅m2⋅s−3⋅A−2\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-2}
Resistance is defined by Ohm's Law as voltage divided by current (R=V/IR = V / I). Voltage is work done per unit charge (V=W/QV = W / Q), and charge is current multiplied by time (Q=I⋅tQ = I \cdot t). Substituting the unit of work (kg⋅m2⋅s−2\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}) gives R=kg⋅m2⋅s−2A2⋅s=kg⋅m2⋅s−3⋅A−2R = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{A}^2\cdot\text{s}} = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-2}.

Step-by-Step Solution

1
Relate resistance to potential difference and current using Ohm's Law.
R=VIR = \frac{V}{I}
Resistance is defined as the ratio of potential difference to electric current.
2
Express electric potential difference (VV) in terms of work (WW) and charge (QQ), where Q=I⋅tQ = I \cdot t.
V=WI⋅t  ⟹  R=WI2⋅tV = \frac{W}{I \cdot t} \implies R = \frac{W}{I^2 \cdot t}
Electric potential difference is work done per unit charge, and electric charge is current multiplied by time.
3
Break down work (W=Force×distanceW = \text{Force} \times \text{distance}) into base SI units.
Unit of Work = kg⋅m2⋅s−2\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}
Force is mass times acceleration (kg⋅m⋅s−2\text{kg}\cdot\text{m}\cdot\text{s}^{-2}), so work is kg⋅m2⋅s−2\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}.
4
Substitute all base units into the resistance formula.
Unit of R=kg⋅m2⋅s−2A2⋅s=kg⋅m2⋅s−3⋅A−2R = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{A}^2 \cdot \text{s}} = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-2}
Combining powers of base units yields the complete fundamental SI base unit expression for the ohm.

Key Concept

Derivation of SI base units from defining formulas of physical quantities
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Topics

Fundamental and Derived Units

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