This paper discusses the SI units needed for the complete definition of the ohm. (Serway and Beichner 845) defines the ohm this way: \[ 1\ \Omega\ \left(\mathrm{ohm}\right)=1\frac{\mathrm{V} }{\mathrm{A}}=\frac{\mathrm{kg} \cdot\mathrm{m}^2}{\mathrm{A}\cdot\mathrm{C}\cdot\mathrm{s}^2} \]

The Encyclopedia American (1950) definition of 1896: ohm = Electrical resistance to an unvarying current offered by a uniform column of mercury at $0° C, 106.3\ \mathrm{cm}$ long and $1\ \mathrm{mm}$ in cross-section.

Primitive SI Units Needed for the Ohm

(Serway and Beichner 944) When the magnitude of the force per unit length between two long, parallel wires that carry identical currents, and are separated by $1\ \mathrm{m},$ is $2\times{10}^{-7}\ \mathrm{N}/\mathrm{m},$ the current in each wire is defined to be $1\ \mathrm{A}~$(ampere).

(Serway and Beichner 944) When a conductor carries a steady current of $1\ \mathrm{A},$ the quantity of charge that flows through a cross-section of the conductor in $1\ \mathrm{s}$ is $1\ \mathrm{C}$ (coulomb). $1\ \mathrm{C}$ is approximately equal to the charge of $6.24\times{10}^{18}$ electrons or protons.

The meter $(\mathrm{m})$ was redefined as the distance traveled by light in vacuum during a time of $1/299{,}792{,}458$ seconds. In effect, this latest definition establishes that the speed of light in vacuum is precisely $299{,}792{,}458\ \mathrm{m/s}.$

The kilogram $(\mathrm{kg})$ is defined as the mass of a specific platinum-iridium alloy cylinder kept at the International Bureau of Weights and Measures at Sèvres, France.

The second $\left(\mathrm{s}\right)$ is defined as $9{,}192{,}631{,}770$ times the period of vibration of radiation from the cesium-133 atom.

Derived SI Units Needed for the Ohm

These derived units appear in the definition of the ohm. \begin{align*} 1\ \mathrm{V} \text{ (volt) } &=1 \frac{\mathrm{J}}{\mathrm{C}}\\ 1\ \mathrm{N} \text{ (newton) } &=1 \frac{\mathrm{kg}\cdot \mathrm{m}}{\mathrm{s}^2}\\ 1\ \mathrm{J} \text{ (joule) } &=1 \mathrm{N}\cdot \mathrm{m} \end{align*}

Works Cited

Encyclopedia Americana. 1950.

Serway, Raymond A. and Robert J. Beichner. Physics for Scientists and Engineers. 5th Edition. Brooks/Cole, 2000. Print.