Match each physical quantity listed on the left with its corresponding SI unit expressed s… — JAMB UTME

Match each physical quantity listed on the left with its corresponding SI unit expressed strictly in terms of fundamental (base) units on the right.

Electric permittivity of free space ( $\varepsilon_0$ ) $\text{kg}^{-1}\cdot\text{m}^{-3}\cdot\text{s}^4\cdot\text{A}^2$
Magnetic permeability of free space ( $\mu_0$ ) $\text{kg}\cdot\text{m}\cdot\text{s}^{-2}\cdot\text{A}^{-2}$
Thermal conductivity ( $k$ ) $\text{kg}\cdot\text{m}\cdot\text{s}^{-3}\cdot\text{K}^{-1}$
Specific heat capacity ( $c$ ) $\text{m}^2\cdot\text{s}^{-2}\cdot\text{K}^{-1}$

Electric permittivity of free space maps to

\text{kg}^{-1}\cdot\text{m}^{-3}\cdot\text{s}^4\cdot\text{A}^2

, Magnetic permeability of free space maps to

\text{kg}\cdot\text{m}\cdot\text{s}^{-2}\cdot\text{A}^{-2}

, Thermal conductivity maps to

\text{kg}\cdot\text{m}\cdot\text{s}^{-3}\cdot\text{K}^{-1}

, and Specific heat capacity maps to

\text{m}^2\cdot\text{s}^{-2}\cdot\text{K}^{-1}

Each physical quantity is matched to its exact fundamental unit expression obtained by substituting basic formulas into SI base units (

\text{kg}

\text{m}

\text{s}

\text{A}

\text{K}

). Electric permittivity resolves to

\text{kg}^{-1}\cdot\text{m}^{-3}\cdot\text{s}^4\cdot\text{A}^2

, magnetic permeability resolves to

\text{kg}\cdot\text{m}\cdot\text{s}^{-2}\cdot\text{A}^{-2}

, thermal conductivity resolves to

\text{kg}\cdot\text{m}\cdot\text{s}^{-3}\cdot\text{K}^{-1}

, and specific heat capacity resolves to

\text{m}^2\cdot\text{s}^{-2}\cdot\text{K}^{-1}

Derive the base SI units for Electric permittivity of free space (

\varepsilon_0

From Coulomb's Law,

F = \frac{q_1 q_2}{4\pi \varepsilon_0 r^2} \implies \varepsilon_0 = \frac{q_1 q_2}{4\pi F r^2}

. Expressing terms in SI fundamental units: charge

q = \text{A}\cdot\text{s}

, force

F = \text{kg}\cdot\text{m}\cdot\text{s}^{-2}

, distance

r = \text{m}

. Thus,

[\varepsilon_0] = \frac{(\text{A}\cdot\text{s})^2}{(\text{kg}\cdot\text{m}\cdot\text{s}^{-2})\cdot\text{m}^2} = \text{kg}^{-1}\cdot\text{m}^{-3}\cdot\text{s}^4\cdot\text{A}^2

Relating derived electromagnetic quantities to fundamental SI units via governing physical equations.

Derive the base SI units for Magnetic permeability of free space (

\mu_0

From the force per unit length between parallel current-carrying conductors,

\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi r} \implies \mu_0 = \frac{2\pi F r}{I_1 I_2 L}

. Units:

[\mu_0] = \frac{(\text{kg}\cdot\text{m}\cdot\text{s}^{-2})\cdot\text{m}}{\text{A}^2\cdot\text{m}} = \text{kg}\cdot\text{m}\cdot\text{s}^{-2}\cdot\text{A}^{-2}

Using Ampère's force law to solve for magnetic permeability in base units.

Derive the base SI units for Thermal conductivity (

k

From Fourier's Law of Heat Conduction,

\frac{Q}{t} = k A \frac{\Delta T}{\Delta x} \implies k = \frac{Q \cdot \Delta x}{t A \Delta T}

. Heat energy

Q = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}

, time

t = \text{s}

, area

A = \text{m}^2

, thickness

\Delta x = \text{m}

, temperature difference

\Delta T = \text{K}

. Thus,

[k] = \frac{(\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2})\cdot\text{m}}{\text{s}\cdot\text{m}^2\cdot\text{K}} = \text{kg}\cdot\text{m}\cdot\text{s}^{-3}\cdot\text{K}^{-1}

Connecting thermal conduction equations to fundamental mechanics and thermodynamic units.

Derive the base SI units for Specific heat capacity (

c

From

Q = m c \Delta T \implies c = \frac{Q}{m \Delta T}

. Units:

[c] = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{kg}\cdot\text{K}} = \text{m}^2\cdot\text{s}^{-2}\cdot\text{K}^{-1}

Applying the defining equation of heat capacity to reduce the unit to fundamental base units.

Derivation of complex derived SI units from fundamental base units using fundamental physical laws.

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