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Difficulty: MediumFundamental and Derived Units

The derived SI unit of electric capacitance, the farad (F\text{F}), can be expressed in terms of fundamental SI base units as kga⋅mb⋅sc⋅Ad\text{kg}^a \cdot \text{m}^b \cdot \text{s}^c \cdot \text{A}^d. What is the numerical value of the sum of the exponents a+b+c+da + b + c + d?

Answer: 3

Answer

The numerical value of the sum of the exponents a+b+c+da + b + c + d is 3.
The farad (F\text{F}), when resolved into fundamental SI base units, is kg−1⋅m−2⋅s4⋅A2\text{kg}^{-1} \cdot \text{m}^{-2} \cdot \text{s}^4 \cdot \text{A}^2. Adding the exponents yields a=−1a = -1, b=−2b = -2, c=4c = 4, and d=2d = 2, giving a total sum of (−1)+(−2)+4+2=3(-1) + (-2) + 4 + 2 = 3.

Step-by-Step Solution

1
Relate electric capacitance to fundamental physical quantities
Capacitance is defined as C=QVC = \frac{Q}{V}, where QQ is electric charge and VV is electric potential difference.
This fundamental relation connects capacitance to charge and energy per unit charge.
2
Express charge and potential difference in terms of SI base units
Electric charge QQ has base units A⋅s\text{A} \cdot \text{s}. Potential difference V=WorkQV = \frac{\text{Work}}{Q} has base units kg⋅m2⋅s−2A⋅s=kg⋅m2⋅s−3⋅A−1\frac{\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}}{\text{A} \cdot \text{s}} = \text{kg} \cdot \text{m}^2 \cdot \text{s}^{-3} \cdot \text{A}^{-1}.
Work is force times distance (kg⋅m⋅s−2×m=kg⋅m2⋅s−2\text{kg} \cdot \text{m} \cdot \text{s}^{-2} \times \text{m} = \text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}) and electric current is a fundamental base quantity.
3
Determine the base unit representation of the farad
The farad is expressed as A⋅skg⋅m2⋅s−3⋅A−1=kg−1⋅m−2⋅s4⋅A2\frac{\text{A} \cdot \text{s}}{\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-3} \cdot \text{A}^{-1}} = \text{kg}^{-1} \cdot \text{m}^{-2} \cdot \text{s}^4 \cdot \text{A}^2.
Dividing the unit of charge by the unit of potential difference yields the derived base unit expression.
4
Calculate the sum of the exponents
a+b+c+d=(−1)+(−2)+4+2=3a + b + c + d = (-1) + (-2) + 4 + 2 = 3.
Summing the powers corresponding to kilograms, meters, seconds, and amperes gives the final scalar value.

Key Concept

Expressing derived SI units in terms of fundamental SI base units
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Topics

Fundamental and Derived Units

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