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Physics 2

Electric current, Ohm's law, Kirchhoff's laws and the RC circuit

Current and drift velocity, resistivity, Joule heating, sources and Kirchhoff's laws, capacitor charging with proof and the energy balance.

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So far charges have been at rest. When they move there is an electric current, and with currents come circuits. This lesson is the bridge between physics and circuit theory: it defines current and resistance starting from the motion of electrons, derives Ohm’s law and Joule heating, introduces Kirchhoff’s laws and studies the RC circuit.

Electric current

Current is the charge crossing a section of the conductor per unit time:

    \[ I = \frac{dQ}{dt}\qquad[\text{A} = \text{C/s}] \]

By convention its direction is the one in which positive charges would move; in metals electrons move the opposite way. The current density \vec J (A/m²) is the current per unit area.

Drift velocity

If the conductor has n carriers per unit volume, each with charge q, moving on average with velocity v_d, then J = n\,q\,v_d.

Example. A 1 mm² copper wire carrying 1 A. Copper has about 8.5\times 10^{28} free electrons per m³. v_d = \frac{J}{ne} = \frac{10^6}{8.5\times 10^{28}\cdot 1.6\times 10^{-19}} \approx 7\times 10^{-5} m/s: less than a tenth of a millimetre per second. A lamp lights up at once not because electrons race along, but because the field travels along the wire at nearly the speed of light and sets all the electrons in the circuit moving at the same time.

Ohm’s law and resistivity

In many materials current density is proportional to the field: \vec J = \sigma\vec E, with \sigma the conductivity. For a wire of length L and cross-section A, with V = EL and I = JA:

    \[ V = R\,I,\qquad R = \rho\,\frac{L}{A} \]

where \rho = 1/\sigma is the resistivity (Ω·m). Copper has \rho = 1.7\times 10^{-8} Ω·m: one metre of 1 mm² wire has 17 mΩ. A printed circuit trace 0.25 mm wide in 35 µm copper has about 2 mΩ per millimetre, and a 10 cm trace carrying 2 A drops 0.4 V: enough to upset a precision circuit.

The resistivity of metals rises with temperature (for copper about 0.4% per degree), that of semiconductors falls. The first effect is used in platinum temperature sensors (Pt100), the second in NTC thermistors.

Power and Joule heating

A charge dQ crossing a potential difference V gives up energy V\,dQ. The power is

    \[ P = V\,I = R\,I^2 = \frac{V^2}{R} \]

In resistors this energy becomes heat (Joule heating): electrons accelerated by the field collide with the lattice and give up energy. It is the principle of electric heaters and fuses, and the reason power lines work at high voltage: for the same power the current is smaller, and the RI^2 losses fall with its square.

Sources and Kirchhoff’s laws

A source maintains a potential difference by spending another form of energy (chemical in a battery, mechanical in a dynamo). Its electromotive force \mathcal E is the work per unit charge; a real source also has an internal resistance r, and across its terminals you measure V = \mathcal E - rI.

Kirchhoff’s laws let you analyse any circuit:

  1. at nodes: the sum of the currents in equals the sum of the currents out (conservation of charge);
  2. around loops: along any closed path the sum of the voltages is zero (the electrostatic field is conservative).

The rules for connections follow: in series resistances add, in parallel conductances 1/R add.

The RC circuit

A capacitor C charges through a resistance R from a source \mathcal E.

RC circuit with source, switch, resistor and capacitor, and graphs of voltage and current
Left, the circuit; right, after the switch closes, the capacitor voltage (red) rises towards E and the current (dashed blue) falls towards zero. After one time constant τ = RC the voltage has reached 63% of its final value.

Proof of the charging law

By the loop law, \mathcal E = R\,i + v, with v = q/C and i = dq/dt = C\,dv/dt:

    \[ RC\,\frac{dv}{dt} + v = \mathcal E \]

It is a first-order linear equation (Analysis 2 lesson): with the capacitor initially empty,

    \[ v(t) = \mathcal E\left(1 - e^{-t/\tau}\right),\qquad i(t) = \frac{\mathcal E}{R}e^{-t/\tau},\qquad \tau = RC \]

At the first instant the empty capacitor behaves like a short circuit (i = \mathcal E/R); in steady state like an open circuit (i = 0). The time constant \tau measures the speed of the transient: with R = 10 kΩ and C = 100 nF, \tau = 1 ms, and after 5\tau = 5 ms the capacitor is 99.3% charged.

The energy balance

The source supplies a total of \mathcal E\,Q = C\mathcal E^2. The capacitor keeps \frac12 C\mathcal E^2. The other half is dissipated in the resistor, \int_0^\infty R\,i^2\,dt = \frac12 C\mathcal E^2, whatever the value of R. Charging a capacitor from a constant voltage source always wastes half the energy: one reason why digital circuits consume in proportion to CV^2f, and why lowering the supply voltage is the most effective way to cut consumption.

The RC circuit is everywhere: in low-pass and high-pass filters, in microcontroller reset circuits, in timers like the 555 and in push-button debouncing circuits.

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