3.2 Resistance and conductance

Resistance and conductance are two sides of the same coin. Resistance describes how much a material opposes current flow, while conductance describes how easily current passes through. Understanding both gives you the tools to analyze circuits and pick the right materials for any electrical application.

Electrical Properties

Resistance and Conductance

Resistance measures the opposition to electric current flowing through a material. It's measured in ohms (Ω). The higher the resistance, the harder it is for current to get through.

Conductance is the flip side: it measures how easily current flows. It's measured in siemens (S). Higher conductance means current passes through more freely.

These two quantities are inverses of each other. If you know one, you can find the other:

$G = \frac{1}{R}$

where $G$ is conductance in siemens and $R$ is resistance in ohms. So a 10 Ω resistor has a conductance of 0.1 S, and a 0.25 S component has a resistance of 4 Ω.

Resistivity and Conductivity

Resistance depends on both the material and the physical dimensions of the object. Resistivity ($\rho$) strips away the size factor and tells you how resistive the material itself is, regardless of shape. It's measured in ohm-meters (Ω·m).

Some reference values show just how wide the range is:

• Copper: $1.68 \times 10^{-8}$ Ω·m (excellent conductor)
• Silicon: $640$ Ω·m (semiconductor)
• Rubber: $\sim 1 \times 10^{13}$ Ω·m (insulator)

Conductivity ($\sigma$) is the reciprocal of resistivity:

$\sigma = \frac{1}{\rho}$

It's measured in siemens per meter (S/m). A material with high conductivity is a good conductor; a material with low conductivity is a good insulator.

Units of Measurement

Ohm and Siemens

The ohm (Ω), named after Georg Ohm, is the SI unit of resistance. By definition, a conductor has a resistance of 1 Ω when a constant potential difference of 1 volt across it produces a current of 1 ampere.

The siemens (S), named after Ernst Werner von Siemens, is the SI unit of conductance. It equals the reciprocal of an ohm: $1 \text{ S} = \frac{1}{1 \text{ Ω}}$. You'll see siemens used frequently in circuit analysis, especially when working with parallel circuits or admittance calculations.

Fundamental Concepts

Inverse Relationship between Resistance and Conductance

The core relationship is straightforward:

$R = \frac{1}{G} \quad \text{and} \quad G = \frac{1}{R}$

This means doubling the resistance cuts the conductance in half, and vice versa. In practice, this lets you convert freely between the two depending on which is more convenient for the problem you're solving.

Material Properties Affecting Resistance and Conductivity

Three main factors determine how resistive a particular piece of material will be: the material itself, its temperature, and its geometry.

Material composition plays the biggest role. Metals like copper and aluminum have high electron mobility, meaning electrons move through them easily, so resistivity is very low. Insulators like rubber and glass have tightly bound electrons, resulting in extremely high resistivity. Semiconductors like silicon fall in between.

Temperature shifts resistivity in different directions depending on the material type:

• In metals, resistance increases as temperature rises. Higher temperatures cause atoms to vibrate more, which scatters electrons and makes it harder for them to flow.
• In semiconductors, resistance decreases as temperature rises. The added thermal energy excites more electrons into the conduction band, creating more charge carriers.

Geometry of the conductor matters too. A longer wire has more resistance; a thicker wire has less. This relationship is captured by:

$R = \rho \frac{l}{A}$

where $\rho$ is the material's resistivity, $l$ is the conductor's length, and $A$ is its cross-sectional area. Think of it like water flowing through a pipe: a longer, narrower pipe restricts flow more than a short, wide one.