The Timpson Lyceum · University Physics II (2026) – Electricity & Magnetism

Current, Resistance, and Electromotive Force – Lecture 7 (University Physics II)

Overview

This lecture finishes the study of capacitors by treating dielectrics quantitatively, then opens a new chapter on current, resistance, and electromotive force. On the capacitor side it shows how a polarizable material inserted between the plates weakens the internal field, raises the capacitance, and lets us reinterpret the constant $\varepsilon_0$ as the permittivity of empty space. It then explains what electric current actually is at the level of drifting electrons, defines resistivity and resistance, states Ohm's law, and introduces the emf and internal resistance of a real battery — closing with electrical power.

Key Concepts

Dielectrics and the dielectric constant

A dielectric is a non-conducting material placed between capacitor plates. Its atoms polarize, building up bound charge that creates a field $E_i$ opposing the original field $E_0$ .
The resultant field is weaker: $E = E_0 - E_i$ . Defining the dielectric constant $K$ , the field is reduced by that factor:

\begin{aligned} E &= \frac{E_0}{K}, \qquad K \ge 1 \end{aligned}

$K$ measures how well the material polarizes: highly polar materials (e.g. water) have large $K$ ; air is very close to $1$ ; a perfect vacuum is exactly $1$ .
Because the field drops while the plate charge is unchanged, the capacitance rises: $C = K C_0$ . Equivalent ways to find $K$ are the ratios $E_0/E$ , $V_0/V$ , and $C/C_0$ .

Permittivity

Inside a dielectric the field is $E = \sigma / (K\varepsilon_0)$ . Writing $\varepsilon = K\varepsilon_0$ as the permittivity of the material lets every earlier formula keep its form with $\varepsilon_0 \to \varepsilon$ :

\begin{aligned} C &= \frac{K \varepsilon_0 A}{d} = \frac{\varepsilon A}{d}, \qquad u = \tfrac{1}{2}K\varepsilon_0 E^2 = \tfrac{1}{2}\varepsilon E^2 \end{aligned}

This finally gives meaning to $\varepsilon_0$ : it is the permittivity of free space, since $1/(4\pi\varepsilon_0)$ is the Coulomb constant.

Dielectric strength vs. dielectric constant

These are independent properties. Dielectric constant $K$ says how much field the material absorbs; dielectric strength says how large a field it can tolerate before it breaks down and conducts (arcs). Water has a high $K$ but poor dielectric strength, so it is a bad choice between plates.

Gauss's law with a dielectric

Only the free charge on the plates matters for circuits, so Gauss's law is written with $K$ and the enclosed free charge:

\begin{aligned} K \oint E \, dA &= \frac{Q_{\text{free}}}{\varepsilon_0} \end{aligned}

Fixed-charge vs. fixed-voltage rule. When a charged capacitor is disconnected from its source, the charge is fixed; when it stays connected to a source, the voltage is fixed. Everything else follows from $C = Q/V$ .

Current and drift velocity

Free electrons in a metal already move randomly at about $10^6\ \text{m/s}$ , but with no net flow there is no current. An applied field gives them a slow net drift velocity on top of that random motion.
Between collisions (average time $\tau$ ) the field $E$ accelerates a charge $e$ of mass $m$ , giving a drift speed $v_d = \dfrac{eE\tau}{m}$ — typically only millimeters per second. A lamp still lights instantly because the wire is already full of electrons that push one another the moment the field appears.
Current is the rate of charge flow, measured in amperes ( $1\ \text{A} = 1\ \text{C/s}$ ): $I = \dfrac{dq}{dt} = n\,e\,v_d\,A$ , where $n$ is the free-electron density.

Resistivity, conductivity, and resistance

Collecting the material-dependent factors gives the resistivity $\rho$ (and its reciprocal, the conductivity $\sigma = 1/\rho$ ). Low resistivity means a good conductor.
Resistance of a uniform wire grows with length and shrinks with cross-sectional area: $R = \dfrac{\rho L}{A}$ , measured in ohms ( $1\ \Omega = 1\ \text{V/A}$ ).
Resistivity depends on temperature: in metals it rises as they heat up (more collisions); semiconductors conduct better when heated; superconductors have zero resistivity below a critical temperature.
Ohm's law, $V = IR$ , is the statement that $\rho$ (hence $R$ ) is essentially constant — a good approximation for metals near room temperature.

Electromotive force and real batteries

An emf $\mathcal{E}$ (measured in volts, not a force) is whatever maintains a potential difference — a battery's chemistry, a generator, etc. A real source also has an internal resistance $r$ , so its terminal voltage is $V_{ab} = \mathcal{E} - I r$ .
With no current ( $I = 0$ ) the terminal voltage equals the emf — that is the number printed on a battery. Under load the measured voltage drops as the current rises.

Electrical power

The power delivered to (or dissipated in) a circuit element is $P = IV$ , in watts ( $1\ \text{W} = 1\ \text{J/s}$ ). Using Ohm's law it can also be written $P = I^2 R = V^2/R$ .

Worked Example

A 12 V battery with internal resistance $r = 2\ \Omega$ drives a single external resistor $R = 4\ \Omega$ . Find the current, the terminal voltage across the battery, and the power dissipated in $R$ .

Apply the loop rule $\mathcal{E} = I(r + R)$ , then the terminal-voltage and power relations:

\begin{aligned} I &= \frac{\mathcal{E}}{r + R} = \frac{12\ \text{V}}{2\ \Omega + 4\ \Omega} = 2\ \text{A} \\ V_{ab} &= \mathcal{E} - I r = 12\ \text{V} - (2\ \text{A})(2\ \Omega) = 8\ \text{V} \\ P_R &= I^2 R = (2\ \text{A})^2 (4\ \Omega) = 16\ \text{W} \end{aligned}

The battery's chemistry supplies $\mathcal{E} I = 24\ \text{W}$ ; of this, $I^2 r = 8\ \text{W}$ is wasted heating the battery internally, leaving the $16\ \text{W}$ delivered to and dissipated by $R$ .

Summary

A dielectric polarizes and weakens the field by a factor $K$ , so $E = E_0/K$ and $C = K C_0$ ; writing $\varepsilon = K\varepsilon_0$ keeps all capacitor formulas intact and identifies $\varepsilon_0$ as the permittivity of free space.
Dielectric strength (resistance to breakdown) is a separate property from the dielectric constant $K$ .
Current is the net drift of electrons: their drift speed is tiny ( $\sim$ mm/s) even though their random speeds are enormous, and $I = dq/dt = n e v_d A$ .
Resistance is $R = \rho L/A$ ; resistivity and conductivity are reciprocals, and Ohm's law $V = IR$ assumes $\rho$ is constant.
A real source has emf $\mathcal{E}$ and internal resistance $r$ , giving terminal voltage $V_{ab} = \mathcal{E} - Ir$ ; the emf is the open-circuit ( $I=0$ ) reading.
Electrical power is $P = IV = I^2 R = V^2/R$ , measured in watts.