Electric Current, Current Density, and Ohm’s Law: Chapter 27 Study Notes

Electric Current

Definition and Indicators

Electric current is the flow of electric charge through a conductor, typically a metal wire. Although the movement of charge is not directly visible, there are observable indicators of current:

• Deflection of a nearby compass needle due to the magnetic field generated by the current.

• Warming of the wire as electrical energy is dissipated as heat.

Current (I) is measured in amperes (A), where 1 ampere equals 1 coulomb of charge passing a point per second ($1~A = 1~C/s$).

Model of Conduction

How Current Flows

Connecting a wire to a battery creates a nonuniform surface charge distribution, which in turn establishes an electric field inside the wire. This field pushes the sea of electrons through the metal, resulting in current.

• Electrons are the charge carriers in metals, but by convention, current is considered as the flow of positive charges.

• Current density (J) is defined as the current per unit area: $J = \frac{I}{A}$, with units of $A/m^2$.

Kirchhoff’s Junction Law

Conservation of Current

Current in a circuit is governed by Kirchhoff’s junction law:

• In a circuit with no junctions, the current is the same everywhere.

• At a junction, the sum of currents entering equals the sum of currents leaving: $\sum I_{in} = \sum I_{out}$.

Resistivity and Resistance

Material and Geometric Properties

Collisions between electrons and atoms in a conductor cause resistance to the motion of charges.

• Resistivity (\(\rho\)) is an intrinsic property of a material, such as copper or aluminum.

• Resistance (R) depends on both the material and the geometry (length and cross-sectional area) of the wire or circuit element.

Ohm’s Law

Relationship Between Current, Voltage, and Resistance

Ohm’s law states that the current through a wire or circuit element is proportional to the potential difference across it and inversely proportional to its resistance:

• $I = \frac{\Delta V}{R}$

Ohm’s law applies to ohmic materials, where resistance remains constant during use.

Charge Carriers in Metals

Sea of Electrons

In metals, outer electrons are weakly bound and can move freely, forming a sea of electrons that acts as the charge carrier.

• Each metal atom typically contributes one conduction electron.

Electron Density in Metals

Tabular Data: Conduction-Electron Density

The number of conduction electrons per cubic meter is approximately equal to the number of atoms per cubic meter.

Metal	Electron density (m-3)
Aluminum	1.8 × 1029
Iron	1.7 × 1029
Copper	8.5 × 1028
Gold	5.9 × 1028
Silver	5.8 × 1028

Discharging a Capacitor

Mechanism and Speed

When a capacitor discharges, the wire is already full of electrons. The discharge occurs rapidly because only a slight rearrangement of charges is needed, not the movement of electrons all the way from one plate to another.

• Typical electron drift speed is $v_d \approx 10^{-4}$ m/s, but discharge is much faster due to the abundance of electrons.

Creating and Sustaining Current

Role of Electric Field

Just as a book requires a continuous push to maintain motion, the sea of electrons in a wire requires a continuous electric field to sustain current against resistive forces.

Establishing the Electric Field in a Wire

Surface Charge Distribution

Connecting wires to a charged capacitor creates a nonuniform surface charge distribution, which establishes an internal electric field that drives electron current from the negative to the positive end.

Model of Conduction

Electron Motion and Drift Speed

• In electrostatic equilibrium, electrons have zero average velocity due to random collisions.

• In an electric field, electrons follow parabolic trajectories between collisions, resulting in a slow net drift (drift speed).

• The mean time between collisions ($\tau$) and electron density ($n_e$) are key properties.

Current and Current Density

Definitions and Formulas

• Current (I): $I = \frac{dQ}{dt}$, where $Q$ is charge.

• Current density (J): $J = \frac{I}{A}$, with units $A/m^2$.

• Direction of current is opposite to electron flow in metals.

Conservation of Current

Charge Conservation Principle

• Current is the same at all points in a wire due to conservation of charge: $I = \text{constant}$.

• In a circuit, the rate of electrons entering and leaving any device is equal.

Kirchhoff’s Junction Law (Detailed)

Mathematical Statement

• At any junction: $\sum I_{in} = \sum I_{out}$

• This law ensures the conservation of electric charge in circuits.

Summary Table: Key Quantities and Units

Quantity	Symbol	Unit
Current	I	A (ampere)
Current density	J	A/m2
Potential difference	ΔV	V (volt)
Resistance	R	Ω (ohm)
Resistivity	ρ	Ω·m

Example: Calculating Current

• If 120 C of charge flows through a wire in one minute, the current is $I = \frac{120~C}{60~s} = 2~A$.

Example: Electron Drift Speed

• Given a current $I$ and wire cross-sectional area $A$, current density is $J = \frac{I}{A}$.

• Electron drift speed $v_d$ can be found using $J = n_e e v_d$.

Key Concepts Review

• Current is the rate of charge flow.

• Current density is current per unit area.

• Ohm’s law relates current, voltage, and resistance.

• Kirchhoff’s junction law ensures conservation of current at circuit junctions.

• Resistivity and resistance describe how materials and components oppose current.