Resistance

Resistance $R$ is the ratio of voltage applied across a conducting region to the steady current that flows through it:

$R=\frac{V}{I}\text{(Ω, ohm)}.$

One ohm is one volt per amp. For a linear (“ohmic”) material, $R$ is a constant of the material and geometry — independent of $V$ and $I$ separately. This is the macroscopic form of Ohm’s law; the microscopic form is the point relation $\mathbf{J}=\sigma \mathbf{E}$.

From the field-level definition

For a uniform conductor of cross-section $S$, length $l$, conductivity $\sigma$, with uniform applied field $E$:

• Current density: $J=\sigma E$.
• Total current: $I=JS=\sigma ES$.
• Voltage drop along the length: $V=El$.
• Resistance: $R=V/I=l/(\sigma S)$.

So

$R=\frac{l}{\sigma S}=\rho \frac{l}{S},$

where $\rho =1/\sigma$ is the resistivity (Ω·m), the material parameter. Long-and-thin conductors have high resistance; short-and-fat conductors have low resistance.

The general integral formula

For non-uniform geometry — e.g., a wedge, a tapered wire, a region with non-uniform $\sigma$ — the relation

$R=\frac{V}{I}=\frac{-{\int }_a^b\mathbf{E}\cdot d\mathbf{l}}{{\int }_S\sigma \mathbf{E}\cdot d\mathbf{s}}\text{(Ω)}$

works for arbitrary geometry. The numerator is the potential difference between the two terminals, computed along any path; the denominator is the total current through any cross-section.

Procedure:

1. Solve Laplace’s equation ${\nabla }^{2}V=0$ in the conductor with boundary conditions $V_a$ and $V_b$ at the two terminals.
2. Compute $\mathbf{E}=-\nabla V$.
3. Apply the integral above.

In simple symmetric cases (uniform wire, coaxial leakage, sphere-in-sphere) this is tractable analytically. For complex shapes you typically simulate (FEM) or measure.

RC product

For any two-conductor geometry with the dielectric having both Permittivity $ϵ$ and conductivity $\sigma$, the product of resistance through the dielectric and the Capacitance of the structure is geometry-independent:

$RC=\frac{ϵ}{\sigma }.$

Proof sketch: both $R$ and $C$ depend on the same $\mathbf{E}$ pattern (determined by the conductor geometry and ${\nabla }^{2}V=0$). The integral for $R$ has $\sigma \mathbf{E}$ in the denominator; the integral for $C$ has $ϵ\mathbf{E}$ in the numerator. Their product collapses to $ϵ/\sigma$ — a material-only constant.

This is useful: compute $C$ from electrostatics (a clean problem), then $R=ϵ/(\sigma C)$ gives the leakage resistance for free. Or measure $R$ and predict $C$.

The same ratio $ϵ/\sigma$ is the dielectric relaxation time $\tau$ — the time scale on which free charge inside a conductor relaxes to the surface. For copper, $\tau \sim 10^{-19}$ s. For mica (a near-perfect insulator), $\tau \sim 15$ hours.

In the lumped-circuit picture

A resistor is a two-terminal lumped element with $V=IR$ (for DC) or, in phasor form, $\tilde{V}=R\tilde{I}$ (frequency-independent for an ideal resistor). It dissipates power:

$P=VI=I^{2}R=V^2/R\text{(W)}.$

This is the macroscopic version of Joule’s law $P/\text{vol}=\mathbf{E}\cdot \mathbf{J}=\sigma E^2$.

For AC at high frequency, real resistors deviate from this ideal — parasitic capacitance and inductance show up in series and parallel. Wire-wound power resistors are inductive; carbon-composition resistors are mildly capacitive. At RF, an “RF resistor” is engineered to minimize these parasitics.

Skin-effect resistance

The DC resistance formula $R=l/(\sigma S)$ assumes current flows uniformly through the cross-section. At high frequency, current is confined to a thin layer of thickness $\delta$ (the skin depth, $\delta =\sqrt{2/(\omega \mu \sigma )}$) near the conductor surface. Effective cross-section drops, and AC resistance rises:

$R_{\text{ac}}\approx \frac{R_s}{\text{perimeter}}\cdot l,R_s=\sqrt{\frac{\omega \mu }{2\sigma }}.$

$R_s$ is the surface resistance (Ω per square). This is the $R^{\prime }$ parameter that appears in the Transmission line lumped model — for a coax it’s $R^{\prime }=R_s/(2\pi )(1/a+1/b)$.

At 1 GHz in copper, $\delta \approx 2$ μm. A wire of 1 mm diameter has nearly all its cross-section unused — drastically more resistive than its DC value. This is why high-frequency conductors are often plated (silver-flashed) or made hollow (waveguide).

Versus impedance

Resistance is the real, dissipative part of impedance $Z=R+jX$. A pure resistor stores no energy; a reactor ($X\ne 0$) stores and returns energy each cycle. In DC steady state, only $R$ remains — capacitors are open, inductors are shorts, leaving only resistive paths to set current levels. In AC steady state, both pieces matter.

In context

Resistance is foundational across every EE subdiscipline:

• Circuit theory: $V=IR$ is the most-used equation in introductory courses.
• Power systems: line resistance sets $I^{2}R$ losses; this is why power transmission is at high voltage (lower $I$ for the same delivered power).
• Signal integrity: the $R^{\prime }$ in the Transmission line lumped model is the conductor resistance per length, contributing to the attenuation constant $\alpha$.
• Sensors: thermistors, strain gauges, photoresistors translate a physical quantity into a resistance change.
• Semiconductors: resistivity varies over orders of magnitude with doping, the basis of every IC.