Theory

Electric current and direct-current circuits

What current and current density are, how Ohm's law and resistivity relate current and voltage, how much power a resistor dissipates, how to solve a circuit with real sources, resistors in series and in parallel and Kirchhoff's laws, and how capacitors charge and discharge in an RC circuit.

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Electric current and current density
Ohm’s law and resistance

Electric current and current density

Throughout this topic the conductors are metallic and homogeneous and the connecting wires have negligible resistance. The currents are steady, except in the RC circuit, where they vary slowly enough that the laws of the steady regime can be applied at every instant.

1. Electric current

An ordered motion of charges is called an electric current. If the charge $Δ q$ passes through a cross section of a conductor in the time interval $Δ t$ , the average current is $\frac{Δ q}{Δ t}$ ; the instantaneous current is

i (t) = \frac{d q}{d t},

and when it does not vary in time it is denoted by $I$ . In the International System it is measured in amperes, $1 A = 1 C/s$ . The conventional direction of the current is the one in which positive charges move; in metals the charge carriers are electrons, which move in the opposite direction.

From the definition it follows that the charge crossing the section between the instants $t_{1}$ and $t_{2}$ is $q = \int_{t_{1}}^{t_{2}} i (t) d t$ , that is, the area under the graph of $i (t)$ . A constant current of $3.0 A$ carries in one minute the charge $3.0 \cdot 60 C = 180 C$ , equal to that of $\frac{180}{1.60 \cdot 1 0 ^{- 19}} \approx 1.1 \cdot 1 0^{21}$ electrons.

2. Current density

The current density $J$ is the vector that at every point of the conductor has the direction of motion of the positive charges and as magnitude the current per unit area through a small surface perpendicular to that direction. The current through a cross section $S$ is the flux of $J$ :

I = \iint_{S} J \cdot \overset{n}{^} d A .

If $J$ is uniform and perpendicular to the cross section of area $A$ , the flux reduces to $I = J A$ . The unit is $A/ m^{2}$ . The current describes the conductor as a whole, the current density describes the motion of the charges point by point; for example, a current of $2.0 A$ in a wire of cross section $1.0 m m^{2}$ has density $J = \frac{2.0}{1.0 \cdot 1 0 ^{- 6}} A/ m^{2} = 2.0 \cdot 1 0^{6} A/ m^{2}$ .

3. Current density and drift velocity

Consider a conductor with $n$ charge carriers per unit volume, each with charge $q$ , moving with average velocity $v_{d}$ , called the drift velocity. Then

J = n q v_{d} .

In metals $q = - e$ , therefore $J$ and $v_{d}$ have opposite directions, and in a wire of cross section $A$ we have $v_{d} = \frac{I}{n e A}$ . The drift velocity is very small: in a copper wire, with $n = 8.5 \cdot 1 0^{28} m^{- 3}$ , cross section $1.0 m m^{2}$ and current $2.0 A$ , it is

v_{d} = \frac{2.0}{8.5 \cdot 1 0 ^{28} \cdot 1.60 \cdot 1 0 ^{- 19} \cdot 1.0 \cdot 1 0 ^{- 6}} m/s \approx 1.5 \cdot 1 0^{- 4} m/s .

A lamp nevertheless lights up as soon as the switch is closed, because the electric field is established along the whole wire in a very short time and sets the electrons of every stretch of the circuit in motion almost simultaneously.

In the time

d t

the carriers contained in the colored cylinder, of length

v_{d} d t

, cross the section

d A

; the arrow shows the direction of the drift velocity

v_{d}

Proof

Consider a surface element $d A$ perpendicular to $v_{d}$ . In the time $d t$ each carrier advances by $v_{d} d t$ , therefore the carriers contained in the cylinder of base $d A$ and height $v_{d} d t$ cross $d A$ . The cylinder has volume $v_{d} d t d A$ and contains $n v_{d} d t d A$ carriers, hence the charge crossing $d A$ is $d q = n q v_{d} d t d A$ . Dividing by $d t$ and by $d A$ gives the magnitude $∣ J ∣ = n ∣ q ∣ v_{d}$ . If $q > 0$ the current has the direction of $v_{d}$ , if $q < 0$ it has the opposite direction; in both cases $J = n q v_{d}$ .

Steady current and conservation of charge

In the steady regime the charge distribution does not change in time, therefore as much charge enters each region of the conductor as leaves it. By conservation of charge, the flux of $J$ through every closed surface is zero:

\iint J \cdot \overset{n}{^} d A = 0.

It follows that in a wire without branches the current is the same in every cross section, even if the cross section changes; where the wire narrows, the current density and the drift velocity increase. By the divergence theorem the relation is equivalent to the local form $\nabla \cdot J = 0$ .

Ohm’s law and resistance

1. Electrical resistance

Consider a conductor with two ends $A$ and $B$ , carrying the current $I$ that enters at $A$ and leaves at $B$ , and let $Δ V = V_{A} - V_{B}$ . The resistance of the conductor is

R = \frac{Δ V}{I},

and it is measured in ohms, $1 Ω = 1 V/A$ . In conductors the current goes from the point at higher potential to the point at lower potential, therefore $R > 0$ . A component designed to have a known resistance is called a resistor.

2. Ohm's law

This is an experimental law: for a metallic conductor at constant temperature the current is proportional to the applied potential difference,

Δ V = R I,

with $R$ independent of $Δ V$ and of $I$ . Conductors that satisfy the law are called ohmic, and their characteristic curve, the graph of $I$ as a function of $Δ V$ , is a straight line through the origin with slope $\frac{1}{R}$ . The law makes it possible to obtain one of the three quantities when the other two are known: a $20 Ω$ resistor connected to $12 V$ carries $I = \frac{12}{20} A = 0.60 A$ .

R_1 = 2\,\Omega

R_2 = 4\,\Omega

Characteristic curves of two ohmic resistors, with

Δ V

in volts and

I

in amperes: they are straight lines through the origin with slope

\frac{1}{R}

, and for the same voltage the resistor with the smaller resistance carries a larger current

Warning

The ratio $\frac{Δ V}{I}$ can be computed for any component, but Ohm’s law holds only if the ratio is constant. A semiconductor diode is a non-ohmic component: with the voltage applied in one direction it carries a negligible current, in the opposite direction the current grows very rapidly with the voltage, and its characteristic curve is not a straight line.

3. Ohm's law in local form

This is an experimental law: in an ohmic homogeneous material the current density is proportional to the electric field,

E = ρ J,

where $ρ$ is the resistivity of the material; in this topic $ρ$ always denotes resistivity, because the volume charge density does not appear. Resistivity is measured in $Ω \cdot m$ , and its reciprocal $σ = \frac{1}{ρ}$ is called conductivity. Resistance is a property of the individual object, resistivity a property of the material. The law gives the electric field inside a wire carrying current: in copper, with $ρ = 1.7 \cdot 1 0^{- 8} Ω \cdot m$ and $J = 2.0 \cdot 1 0^{6} A/ m^{2}$ , we have $E = 0.034 V/m$ . The field is nonzero because the conductor is not in electrostatic equilibrium; for conductors in equilibrium see the topic on conductors, capacitors and dielectrics.

4. Resistance of a wire

Consider a homogeneous cylindrical conductor of length $ℓ$ , cross-sectional area $A$ and resistivity $ρ$ , carrying a steady current with uniform density parallel to the axis. Then

R = ρ \frac{ℓ}{A} .

The resistance is proportional to the length and inversely proportional to the cross-sectional area. A copper wire $10 m$ long with a cross section of $1.0 m m^{2}$ has $R = 1.7 \cdot 1 0^{- 8} \cdot \frac{10}{1.0 \cdot 1 0 ^{- 6}} Ω = 0.17 Ω$ , a resistance negligible compared with that of almost all the appliances it supplies.

Proof

The current density is uniform and perpendicular to the cross section, therefore $J = \frac{I}{A}$ . By Ohm’s law in local form the electric field is uniform, parallel to the axis and of magnitude $E = ρ J = \frac{ρ I}{A}$ . The potential difference between the ends is the line integral of the field along the axis, and for a uniform field parallel to the path it is

Δ V = \int_{0}^{ℓ} E d x = E ℓ = \frac{ρ ℓ}{A} I .

Hence $R = \frac{Δ V}{I} = ρ \frac{ℓ}{A}$ .

The theory continues with 6 more sections (resistivity and temperature, power and the joule effect, sources and electromotive force…), plus all worked examples, the solutions and the PDF.

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