= Electrical resistance
{title2=$R$}
<Electrical resistance> measures the <electric potential difference> needed to drive a given <electric current> through a component. For an <Ohmic conductor> under fixed material conditions, $R=V/I$ is constant and <Ohm's law> gives $V=IR$. The <SI> unit is the <ohm>, $1\,\Omega=1\,\mathrm{V/A}$. At a fixed positive <electric potential difference>, increasing <electrical resistance> reduces <electric current>. <Electrical conductance> $G=1/R$ expresses the same response reciprocally. https://openstax.org/books/university-physics-volume-2/pages/9-3-resistivity-and-resistance[Ling, Sanny and Moebs, §9.3: resistance and its unit]{ref}.
For a straight uniform wire at rest, with constant <electrical conductivity> $\sigma>0$, <length> $L$ and <wire cross-sectional area> $A$, the uniform <Ohmic conductor> model gives
$$
R=\frac{L}{\sigma A}=\rho_{\mathrm{el}}\frac LA,\qquad \rho_{\mathrm{el}}=\frac1\sigma.
$$
The <electrical resistivity> $\rho_{\mathrm{el}}$ characterizes the material; <electrical resistance> also includes geometry. Doubling <length> doubles $R$, while doubling <wire cross-sectional area> halves $R$, with material and <temperature> fixed. The uniform-field assumption includes insulated sides and equipotential end faces, as explained in <uniform current in a straight homogeneous wire>. https://openstax.org/books/university-physics-volume-2/pages/9-3-resistivity-and-resistance[Ling, Sanny and Moebs, §9.3: wire geometry and resistivity]{ref}.
As a defined wire example, take $L=1\,\mathrm m$, $A=1\,\mathrm{mm^2}$ and $\sigma=5\times10^5\,(\Omega\,\mathrm m)^{-1}$. Then $R=2\,\Omega$; applying <electric potential difference> $5\,\mathrm V$ drives <electric current> $2.5\,\mathrm A$. With the same <length> and material, a second wire of <wire cross-sectional area> $2\,\mathrm{mm^2}$ has $R=1\,\Omega$ and carries $5\,\mathrm A$ at that same <electric potential difference>. Fixing <electric potential difference> here differs from fixing <electric current> in the <current density> comparison: stating what is held fixed is essential.
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