Current and Resistance: Study Notes (Physics for Scientists and Engineers, Ch. 27)

Current and Resistance

Introduction to Electric Current

Electric current is a fundamental concept in physics, describing the flow of electric charge through a conductor. This chapter explores the nature of current, the mechanisms of conduction, and the laws governing current and resistance in materials.

• Electric Current (I): The rate at which charge (Q) flows through a conductor. Measured in amperes (A), where 1 A = 1 C/s.

• Indicators of Current: Deflection of a nearby compass needle and warming of a current-carrying wire.

• Conventional Current: Defined as the direction positive charges would move, opposite to the actual electron flow in metals.

Charge Carriers and the Sea of Electrons

In metals, the outer electrons are loosely bound and form a 'sea' of conduction electrons, which are free to move throughout the lattice of positive ions.

• Charge Carriers: In metals, electrons are the primary charge carriers.

• Electron Current (i_e): The number of electrons passing through a cross-section per second.

• Electron Density (n_e): Number of conduction electrons per unit volume, typically equal to the number of atoms per cubic meter in metals.

Current, Current Density, and Drift Speed

Current and current density are related to the microscopic motion of electrons in a conductor.

• Current (I): $I = \frac{\Delta Q}{\Delta t}$

• Current Density (J): $J = \frac{I}{A}$, where A is the cross-sectional area (units: A/m2).

• Drift Speed (v_d): The average velocity of electrons due to the electric field, typically very slow (≈ $10^{-4}$ m/s).

• Relation to Electron Current: $I = n_e e v_d A$

Example: Doubling both the wire diameter and the electron drift speed increases the electron current by a factor of 4 (since area increases by 4 and drift speed by 2, total factor is 8).

Establishing Current in a Wire

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

• Surface Charges: Rearranged to create the internal electric field.

• Electric Field Direction: Points from the more positive to the more negative end; electrons drift opposite to this field.

Microscopic Model of Conduction

Electrons in a conductor undergo frequent collisions with the atomic lattice. In the absence of an electric field, their average velocity is zero. When an electric field is applied, electrons acquire a small net drift velocity superimposed on their random thermal motion.

• Mean Time Between Collisions (\tau): Average time between electron collisions.

• Drift Speed Formula: $v_d = \frac{e E \tau}{m_e}$

• Electron Current Formula: $I = n_e e v_d A$

Conservation of Current and Kirchhoff’s Junction Law

Current is conserved in a circuit. The current entering any junction equals the current leaving it, a principle known as Kirchhoff’s junction law.

• Kirchhoff’s Junction Law: $\sum I_{in} = \sum I_{out}$

• Conservation of Charge: Ensures current is the same at all points in a series circuit.

Conductivity and Resistivity

Materials differ in their ability to conduct electric current, characterized by conductivity (\sigma) and resistivity (\rho).

• Conductivity (\sigma): A measure of how easily a material allows current to flow. Units: S/m (siemens per meter).

• Resistivity (\rho): A measure of how strongly a material opposes current. Units: Ω·m. $\rho = \frac{1}{\sigma}$

• Relation to Current Density: $J = \sigma E$ or $E = \rho J$

Example: Measuring body fat by passing a current through the body; muscle and fat have different resistivities, affecting the measured current.

Material	Resistivity (Ω·m)	Conductivity (S/m)
Copper	1.68 × 10-8	5.96 × 107
Aluminum	2.82 × 10-8	3.5 × 107
Nickel-chromium alloy*	1.10 × 10-6	9.1 × 105
Silver	1.59 × 10-8	6.30 × 107
Gold	2.44 × 10-8	4.10 × 107

*Nickel-chromium alloy is commonly used for heating wires.

Resistance and Ohm’s Law

Resistance quantifies how much a conductor opposes the flow of current. Ohm’s law relates current, voltage, and resistance in ohmic materials.

• Resistance (R): $R = \frac{\Delta V}{I}$, measured in ohms (Ω).

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

• Ohm’s Law: $I = \frac{\Delta V}{R}$

• Ohmic Materials: Materials for which resistance remains constant as voltage varies (e.g., metals).

• Nonohmic Materials: Devices like diodes, batteries, and capacitors, where current is not proportional to voltage.

Example: A 9.0 V battery connected to a resistor with a 15 mA current: $R = \frac{9.0\ \text{V}}{0.015\ \text{A}} = 600\ \Omega$

Superconductivity

Superconductivity is a phenomenon where certain materials lose all electrical resistance when cooled below a critical temperature. Superconductors exhibit unique magnetic properties, such as magnetic levitation.

• Critical Temperature: The temperature below which a material becomes superconducting.

• Applications: Magnetic levitation, MRI machines, and lossless power transmission.

Summary Table: Key Quantities and Relationships

Quantity	Symbol	Formula	SI Unit
Current	I	$I = \frac{\Delta Q}{\Delta t}$	A (ampere)
Current Density	J	$J = \frac{I}{A}$	A/m2
Drift Speed	v_d	$v_d = \frac{I}{n_e e A}$	m/s
Resistance	R	$R = \rho \frac{L}{A}$	Ω (ohm)
Ohm’s Law	-	$I = \frac{\Delta V}{R}$	-
Conductivity	\sigma	$J = \sigma E$	S/m
Resistivity	\rho	$E = \rho J$	Ω·m

Applications and Examples

• Resistors: Devices designed to have a specific resistance, used to control current in circuits.

• Batteries: Provide a potential difference (ΔV) that drives current through a circuit.

• Lightbulbs: Convert electrical energy into light and heat; brightness depends on current.

Key Points for Exam Preparation

• Understand the difference between conventional current and electron flow.

• Be able to calculate current, current density, drift speed, resistance, and apply Ohm’s law.

• Know the physical meaning of conductivity and resistivity, and how they relate to material properties.

• Apply Kirchhoff’s junction law to analyze simple circuits.

• Recognize the characteristics of ohmic and nonohmic materials.

• Be familiar with the phenomenon of superconductivity and its implications.

Additional info: Some values and example calculations were inferred based on standard textbook content and the context of the provided notes.