Electric Current, Conductors, and Ohm's Law

Motion of a Point Charge in an Electric Field

Force and Acceleration of a Charged Particle

When a particle of mass m and charge q is placed in an electric field E, it experiences a force and accelerates accordingly.

• Force on a charge: $F = qE$

• Newton's Second Law: $F = ma$

• Acceleration of the particle: $a = \frac{q}{m}E$

• If the electric field is produced by a potential difference V across a distance L: $E = \frac{V}{L}$, so $a = \frac{q}{m}\frac{V}{L}$

Example: An electron in a uniform electric field will accelerate opposite to the field direction due to its negative charge.

Conductors in Electric Fields

Behavior of Charges in Conductors

Conductors, typically metals, contain free electrons (conduction electrons) that are not bound to atoms and can move freely within the material.

• In the absence of an external field: Free electrons move randomly due to thermal energy, resulting in no net current.

• When an electric field is applied: Electrons move and accumulate at the surface, creating an internal field that cancels the applied field inside the conductor (electrostatic equilibrium).

• Surface charge: Negative electrons accumulate on one surface, positive ions on the opposite, screening the field inside.

• Electrons do not leave the conductor: It requires significant energy to remove electrons from the surface (work function).

Example: In a metal rod placed in an electric field, electrons redistribute until the internal field is zero.

Conduction Electrons and Random Motion

Random Motion and Collisions

In the absence of an electric field, conduction electrons undergo random, Brownian-like motion due to frequent collisions with ions in the lattice.

• Random trajectories: No net movement of charge in any direction.

• Collisions: Electrons scatter off ions, changing direction and speed.

Influence of Electric Field on Electron Flow

Drift Motion

When an electric field is applied, the random motion of electrons is superimposed with a slow drift opposite to the field direction.

• Drift velocity: The average velocity of electrons due to the field, much smaller than their random thermal velocity.

• Direction: Electrons drift opposite to the electric field; conventional current is defined in the direction positive charges would move.

Example: In a copper wire, electrons drift toward the positive terminal when a voltage is applied.

Model of Conduction (Drude Model)

Electron Collisions and Drift Velocity

The Drude model treats conduction electrons as classical particles that accelerate between collisions with ions.

• Mean time between collisions: $\tau$

• Drift velocity: $v_d = \frac{e\tau}{m}E$

• Average velocity: The net drift is much smaller than the random speed between collisions.

Example: For copper, the drift velocity is typically less than 1 mm/s, while the random speed is on the order of $10^5$ m/s.

Electric Current

Definition and Direction

Electric current is the rate at which charge flows through a surface.

• Current: $I = \frac{dq}{dt}$

• Average current: $I_{avg} = \frac{\Delta Q}{\Delta t}$

• Unit: 1 Ampere (A) = 1 Coulomb/second (C/s)

• Direction: Defined as the direction positive charges would move (conventional current); electrons move in the opposite direction.

Example: If 120 C of charge pass through a wire in 1 minute, the current is $I = \frac{120\,\text{C}}{60\,\text{s}} = 2\,\text{A}$.

Current Density

Definition and Calculation

Current density describes how much current flows per unit area of a conductor.

• Current density: $J = \frac{I}{A}$

• Units: A/m2

Example: For a wire of 2 mm diameter carrying 2 A, $A = \pi r^2 = \pi (0.001\,\text{m})^2 = 3.14 \times 10^{-6}\,\text{m}^2$, so $J = \frac{2}{3.14 \times 10^{-6}} \approx 6.4 \times 10^5\,\text{A/m}^2$.

Conservation of Current

Current in Series Segments

In a steady-state circuit, the current entering a segment of wire equals the current leaving it, provided there is no accumulation of charge.

• Conservation law: $I_1 = I_2$ for segments in series made of the same material.

• Current density: If the cross-sectional area changes, $J = \frac{I}{A}$, so $J$ increases as $A$ decreases for constant $I$.

Example: If a wire narrows, the current remains the same, but the current density increases in the narrower section.

Microscopic Model of Current (Drude Model)

Drift Velocity and Current

The current in a conductor can be related to the drift velocity of electrons and the number of charge carriers.

• Number density of electrons: $n$ (number per unit volume)

• Drift velocity: $v_d$

• Current: $I = n e v_d A$

• Current density: $J = n e v_d$

Example: For copper with $n = 8.49 \times 10^{28}\,\text{m}^{-3}$, $I = n e v_d A$ can be used to find $v_d$ for a given current and wire size.

Ohm's Law

Microscopic and Macroscopic Forms

Ohm's Law relates current, voltage, and resistance in a conductor.

• Microscopic form: $J = \sigma E$, where $\sigma$ is the conductivity ($\sigma = \frac{n e^2 \tau}{m}$)

• Resistivity: $\rho = \frac{1}{\sigma}$

• Macroscopic form: $V = IR$

• Resistance: $R = \rho \frac{L}{A}$, where $L$ is length and $A$ is cross-sectional area

• Units: Resistance in ohms ($\Omega$), resistivity in $\Omega\cdot$m

Example: For a Nichrome wire of radius 2.00 mm, required to draw 8.50 A at 120 V, the resistance needed is $R = \frac{V}{I}$, and the length can be found from $R = \rho \frac{L}{A}$.

Table: Resistivity of Selected Materials

The resistivity of materials varies widely and determines their suitability as conductors or insulators.

Material	Resistivity ($\Omega\cdot$m)	Temperature Coefficient
Copper	1.68 × 10-8	0.0039
Aluminum	2.82 × 10-8	0.0043
Gold	2.44 × 10-8	0.0034
Nichrome	1.50 × 10-6	0.0004
Glass	1010–1014	—
Rubber	1013	—

Additional info: Table values are typical and may vary by source and temperature.