3.3 Electric Current, Resistance & Resistor Networks

3.3 Electric Current, Resistance & Resistor Networks

Electric circuits describe the closed pathways through which electric charges flow under the influence of potential differences. Mastery of circuit theory requires understanding charge conservation, energy conservation, network reduction, and transient charging dynamics.

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1. Electric Current, Drift Velocity, and Ohm's Law

Electric current ($I$) quantifies the net rate of electric charge transport through a defined cross-sectional boundary:

$$I = \frac{\Delta Q}{\Delta t}$$

• Units: Amperes ($\text{A}$), where $1\text{ A} = 1\text{ Coulomb per second}$ ($1\text{ C}/\text{s}$).
• Conventional Current vs. Electron Drift: By historical convention established by Benjamin Franklin, current is defined as the direction in which positive charge carriers would flow—from higher electrical potential ($+$) toward lower electrical potential ($-$). In metallic conductors, mobile charge carriers are negative conduction electrons drifting in the opposite direction.

Ohm's Law

Ohm's law states that for an ohmic conductor, the potential difference ($V$) across the component is directly proportional to the current ($I$) passing through it:

$$V = I R$$

Where $R$ is the electrical resistance, measured in Ohms ($\Omega = \text{V}/\text{A}$):

• Ohmic Conductors: Exhibit a linear current-voltage relationship ($I$ vs. $V$ is a straight line through the origin with slope $1/R$).
• Non-Ohmic Devices: Exhibit nonlinear $I$-$V$ curves where resistance varies with applied voltage or current. Biological voltage-gated ion channels, semiconductor diodes, and action potential thresholds are distinctly non-ohmic.

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2. Resistance, Resistivity, and Geometric Dependencies

Resistance is an extrinsic property that depends on both the intrinsic material composition and the physical geometry of the conducting pathway:

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

Where:

• $\rho$ is the resistivity of the material (in $\Omega\cdot\text{m}$), reflecting intrinsic resistance to electron passage. Conductivity ($\sigma$) is the reciprocal of resistivity ($\sigma = 1/\rho$, in $(\Omega\cdot\text{m})^{-1}$ or $\text{S}/\text{m}$).
• $L$ is the length of the conductor (in meters). Resistance is directly proportional to length ($R \propto L$); doubling length doubles resistance.
• $A$ is the cross-sectional area (in $\text{m}^2$). Resistance is inversely proportional to area ($R \propto 1/A$).

Cylindrical Geometries in Biology

For cylindrical conductors such as blood vessels or neuronal axons with radius $r$ (or diameter $d = 2r$):

$$A = \pi r^2 = \frac{\pi d^2}{4} \implies R = \frac{\rho L}{\pi r^2}$$

Temperature Dependence of Resistance

In metallic conductors, electrical resistivity increases with rising temperature ($\rho(T) = \rho_0[1 + \alpha(T - T_0)]$) because thermal lattice vibrations increase the frequency of scattering collisions between conduction electrons and lattice ions.

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3. Resistor Networks and Kirchhoff's Laws

Complex circuits can be resolved into equivalent resistances ($R_{\text{eq}}$) using series and parallel combination rules.

Resistors in Series

When resistors are connected end-to-end along a single conducting branch:

$$R_{\text{eq}} = R_1 + R_2 + R_3 + \dots + R_n$$

1. Current is Identical: The same current must flow sequentially through every resistor ($I_{\text{total}} = I_1 = I_2 = I_3$).
2. Voltage Divides Additively: The total potential drop across the series chain is the sum of the individual voltage drops ($V_{\text{total}} = V_1 + V_2 + V_3$).
3. Equivalent Resistance Increases: $R_{\text{eq}}$ is always strictly greater than the largest individual resistor in the series combination.

Resistors in Parallel

When resistors are connected across the same two electrical nodes:

$$\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_n}$$

For exactly two resistors in parallel, this simplifies to the "product over sum" formula:

$$R_{\text{eq}} = \frac{R_1 R_2}{R_1 + R_2}$$

1. Voltage is Identical: The potential difference across each parallel branch is identical ($V_{\text{total}} = V_1 = V_2 = V_3$).
2. Current Divides Inversely: Total current splits among branches, with lower-resistance paths carrying proportionally more current ($I_{\text{total}} = I_1 + I_2 + I_3$).
3. Equivalent Resistance Decreases: Adding another parallel branch provides an additional conductive conduit, always decreasing total $R_{\text{eq}}$. $R_{\text{eq}}$ is always strictly less than the smallest individual resistor in the parallel group.

Kirchhoff's Circuit Laws

1. Kirchhoff's Junction Rule (Current Law): The sum of currents entering any electrical junction must equal the sum of currents leaving that junction: $$\sum I_{\text{in}} = \sum I_{\text{out}}$$ Physical Basis: Conservation of Electric Charge.
2. Kirchhoff's Loop Rule (Voltage Law): The algebraic sum of all potential differences around any closed circuit loop is zero: $$\sum \Delta V = 0$$ Physical Basis: Conservation of Energy.

Electrical Power and Battery Internal Resistance

Electrical power ($P$) dissipated as heat across a resistor or delivered by a source is:

$$P = I V = I^2 R = \frac{V^2}{R}$$

Real non-ideal voltage sources possess an internal resistance ($r_{\text{int}}$). The terminal voltage ($V_{\text{terminal}}$) delivered to an external load resistance ($R_{\text{load}}$) is reduced by the internal voltage drop ($I r_{\text{int}}$):

$$V_{\text{terminal}} = \mathcal{E} - I r_{\text{int}}$$

Where $\mathcal{E}$ is the electromotive force (ideal open-circuit voltage).

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Series versus Parallel Resistor Behaviour

Passage questions rarely ask for a numerical equivalent resistance without also testing which quantity is shared across the elements. Memorise the sharing rule, not just the formula.