Circuit Elements, Resistance and Resistivity

Tip

Exam Focus

Bread-and-butter marks: Ohm’s law $V=IR$ and its limitations, the power trio $P=VI=I^2R=V^2/R$, resistivity $R=\rho\ell/A$ with the temperature law $R_T=R_0(1+\alpha\,\Delta T)$, series/parallel reduction, and the voltage- and current-divider rules. Every numerical in this chapter builds on these.

Ohm’s Law

Note

Law — Ohm’s law

At constant temperature (and other physical conditions held fixed), the current through a conductor is directly proportional to the potential difference across it:

$$\boxed{V=IR},\qquad [V]=[A]\times[\Omega].$$

Limitations. Ohm’s law does not hold for non-linear devices (diode, transistor, thermistor), for gas/arc conductors, or whenever the temperature changes appreciably — their $V$–$I$ characteristic is not a straight line.

Caution

Key Formula — Power in a resistor

$$P=VI=I^2R=\frac{V^2}{R}\quad[W].$$

Resistance, Resistivity and Temperature

Caution

Key Formula — Resistivity, conductance, conductivity

$$R=\rho\,\frac{\ell}{A},\qquad G=\frac1R\ [S],\qquad \sigma=\frac1\rho\ [S/m],$$

where $\rho$ is the resistivity ($\Omega m$), $\ell$ the length and $A$ the cross-sectional area.

For metallic conductors the resistance rises almost linearly with temperature:

Caution

Key Formula — Temperature coefficient of resistance

$$R_T=R_0\bigl(1+\alpha\,\Delta T\bigr),$$

$\alpha$ = temperature coefficient ($/K$); $\alpha>0$ for metals, $\alpha<0$ for carbon/semiconductors.

The Three Passive Elements

Element	$V$–$I$ relation	Stores/dissipates	Energy
Resistor $R$	$v=Ri$	Dissipates as heat	$P=I^2R$
Inductor $L$	$v=L\,\dfrac{di}{dt}$	Magnetic field	$W=\tfrac12 Li^2$
Capacitor $C$	$i=C\,\dfrac{dv}{dt}$	Electric field	$W=\tfrac12 Cv^2$

The resistor, inductor and capacitor and their energy behaviour.

Series and Parallel Combinations

Caution

Key Formula — Series and parallel resistance

$$R_{\text{series}}=R_1+R_2+\cdots,\qquad \frac{1}{R_{\text{parallel}}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots$$

Series and parallel resistor networks.

Caution

Key Formula — Divider rules

$$\text{Voltage divider: } V_1=V\,\frac{R_1}{R_1+R_2},\qquad \text{Current divider: } I_1=I\,\frac{R_2}{R_1+R_2}.$$

[!NOTE]

Example — Series–parallel reduction

A 12 V source feeds $R_1=4\,\Omega$ in series with the parallel combination of $R_2=6\,\Omega$ and $R_3=3\,\Omega$. Find the source current.
$R_{23}=\dfrac{6\times3}{6+3}=2\,\Omega$, so $R_{\text{tot}}=4+2=6\,\Omega$ and $I=12/6=2\,A$.