Current and Resistance

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 🌐 Version en français : Courant et résistance

📘 Current and Resistance

Reference Textbook

📖 Reference Textbook:
University Physics Volume 2: Chapter 9 — Current and Resistance

Theory

Electric Current

Electric current is the rate at which electric charge passes through a cross section:

$I=\frac{dq}{dt}$

For a constant current, $I=\mathrm{\Delta }q/\mathrm{\Delta }t$. The SI unit is the ampere (A):

$1\mathrm{A}=1\mathrm{C}/\mathrm{s}$

Conventional Current Direction

The conventional current direction is the direction in which positive charges would move. In a metal, the mobile charge carriers are electrons, so their drift is opposite to the conventional current direction.

Inside a resistor, conventional current flows from higher to lower electric potential, in the direction of the electric field.

Current Density and Drift Speed

The current density $\overrightarrow{J}$ describes the current per unit area and points in the direction of conventional current. Its SI unit is $\mathrm{A}/{\mathrm{m}}^{\mathrm{2}}$.

For a uniform current density perpendicular to a cross section of area $A$, the magnitudes satisfy

$J=\frac{I}{A}$

For one type of charge carrier,

$\overrightarrow{J}=nq{\overrightarrow{v}}_{\mathrm{d}}$

where:

• $n$ is the number of mobile charge carriers per unit volume (in ${\mathrm{m}}^{\mathrm{-}\mathrm{3}}$);
• $q$ is the charge of each carrier (in coulombs, C);
• ${\overrightarrow{v}}_{\mathrm{d}}$ is their average drift velocity (in $\mathrm{m}/\mathrm{s}$).

For electrons, $q=-e$, where $e=1.602\times 10^{-19}\mathrm{C}$. In terms of magnitudes,

$J=ne{v}_{\mathrm{d}},I=neA{v}_{\mathrm{d}}$

Electron drift speeds in ordinary wires can be very small, of order $10^{-5}$ to $10^{-4}\mathrm{m}/\mathrm{s}$, depending on the current density and material. This slow average drift is distinct from the much faster propagation of an electrical signal through a circuit.

▶ Video: Current and Current Density — 1

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▶ Video: Current and Current Density — 2

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Conservation of Charge

Electric charge is conserved. Charge does not accumulate in a wire or at a junction. Therefore, the current is the same at every cross section of an unbranched wire, even if its cross-sectional area changes.

At a junction, the total current entering equals the total current leaving:

$\sum I_{\mathrm{i}\mathrm{n}}=\sum I_{\mathrm{o}\mathrm{u}\mathrm{t}}$

Equivalently, $\sum I=0$ when entering and leaving currents are assigned opposite signs. This is Kirchhoff's junction rule, used in DC circuit analysis.

Resistance, Resistivity, and Conductivity

Resistance $R$ describes how strongly a component opposes current. It is the ratio of the voltage across the component to the current through it:

$R=\frac{V}{I}$

Here, $V$ denotes the magnitude of the potential difference across the resistor, and $I$ denotes the current magnitude. The SI unit of resistance is the ohm:

$1\mathrm{\Omega }=1\mathrm{V}/\mathrm{A}$

Resistivity $\rho$ is a material property, measured in $\mathrm{\Omega }\cdot \mathrm{m}$. For a uniform wire of length $L$ and constant cross-sectional area $A$,

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

A longer wire has a larger resistance; a thicker wire has a smaller resistance, for the same material under the same conditions. Resistivity depends on the material and conditions such as temperature.

Conductivity $\sigma$ is the reciprocal of resistivity:

$\sigma =\frac{1}{\rho }$

▶ Video: Resistance, Resistivity, and Conductivity — 1

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▶ Video: Resistance, Resistivity, and Conductivity — 2

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Ohm's Law

An ohmic component has a current proportional to the applied voltage. Its resistance remains constant when physical conditions, particularly temperature, are held constant:

$V=IR$

Its current–voltage graph is a straight line through the origin. Not all components obey Ohm's law: for a non-ohmic component, the ratio $V/I$ may vary with the applied voltage.

For an ohmic material, the microscopic form of Ohm's Law is

$\overrightarrow{E}=\rho \overrightarrow{J},\overrightarrow{J}=\sigma \overrightarrow{E}$

▶ Video: Ohm's Law

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Electrical Power and Energy

As charge passes through a resistor, electric potential energy is converted into thermal energy. If $V$ is the voltage drop and $dq$ is the amount of charge passing through, the energy transferred to the resistor is equal to the negative of the change in electric potential energy: $dU=-Vdq$.

Power is the rate of energy transfer, so here, the magnitude of the power is:

$P=\frac{dU}{dt}=V\frac{dq}{dt}=VI$

The SI unit is the watt (W):

$1\mathrm{W}=1\mathrm{J}/\mathrm{s}$

Using $V=IR$, the power dissipated by a resistor can also be written as

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

For constant power over a time interval $\mathrm{\Delta }t$, the energy transferred is

$Energ{y}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{d}}=P\mathrm{\Delta }t$

What Is a Kilowatt-Hour?

A kilowatt-hour (kWh) is a unit of energy. It is the energy transferred by a device operating at a power of 1 kW for 1 hour:

$1\mathrm{k}\mathrm{W}\mathrm{h}=1000\mathrm{W}\times 3600\mathrm{s}=3.6\times 10^6\mathrm{J}$

For example, a 2 kW heater operating continuously for 3 hours uses 6 kWh of electrical energy.

Current and Resistance Simulations

• PhET: Ohm's Law — Explore the relationship between voltage, current, and resistance.

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