Study Notes: Current, Resistance, DC Circuits, Magnetic Fields, and Electromagnetic Induction

Ch. 27: Current and Resistance

Current: The Motion of Charges

Electric current is the flow of electric charge, typically carried by electrons in a conductor. Understanding current is fundamental to analyzing electrical circuits.

• Current in Conductors: In a conductor, electrons move randomly but drift in response to an electric field, creating a net flow of charge.

• Drift Speed: The average velocity of electrons due to the electric field is called drift speed ($v_d$).

• Electron Current: The rate at which electrons pass through a cross-section is $I_e = n_e v_d A$, where $n_e$ is electron density, $A$ is cross-sectional area.

• Conventional Current: Defined as the flow of positive charge; direction is opposite to electron flow.

• Current Density ($J$): Current per unit area, $J = I/A$.

Key Equations:

• $I = n_e e v_d A$

• $J = n_e e v_d$

Mathematical Definition of Current

• Current ($I$): $I = \frac{dQ}{dt}$, where $Q$ is charge passing through a cross-section in time $dt$.

• SI Unit: Ampere (A), $1\,\text{A} = 1\,\text{C}/\text{s}$.

Kirchhoff's Junction Law

States that the sum of currents entering a junction equals the sum leaving it, reflecting conservation of charge:

• $\sum I_{in} = \sum I_{out}$

Conductivity and Resistivity

• Conductivity ($\sigma$): Measures how easily a material conducts electric current. $J = \sigma E$.

• Resistivity ($\rho$): Measures how strongly a material opposes current. $\rho = 1/\sigma$.

Ohm's Law

Relates voltage, current, and resistance in a conductor:

• $V = IR$

• Resistance for a wire: $R = \rho \frac{L}{A}$

Superconductors

• Materials with zero resistivity below a critical temperature; electrons move without energy loss.

Summary Table of Variables

Variable	Represents	Units	Equation
$n_e$	Number density of electrons	None	$n_e = \frac{N_e}{V}$
$I$	Current	A (Ampere)	$I = n_e e v_d A$
$v_d$	Drift speed	m/s	$v_d = \frac{I}{n_e e A}$
$J$	Current density	A/m2	$J = n_e e v_d$
$\sigma$	Conductivity	S/m	$J = \sigma E$
$\rho$	Resistivity	Ω·m	$\rho = 1/\sigma$
$R$	Resistance	Ω	$R = \rho \frac{L}{A}$

Ch. 28: DC Circuits

DC Circuits Overview

DC (Direct Current) circuits use sources of electric potential energy to maintain a constant current. Analysis involves understanding how current flows and how energy is distributed.

• Battery: Provides electric potential energy.

• Load: Elements that consume electrical energy (resistors, bulbs, etc.).

• Circuit Diagrams: Use standardized symbols to represent components.

Analysis of a Single Loop Circuit

• Current flows in one direction; the sum of potential differences around the loop equals zero.

• For a resistor: $V = IR$

• For a battery: $V_{battery}$

Energy and Power in Circuits

• Power Dissipated: $P = IV = I^2R = \frac{V^2}{R}$

• Energy Consumed: $E = P \times t$

• Kilowatt-hour: Unit of energy used by electric utilities.

Resistors in Series and Parallel

• Series: $R_{eq} = R_1 + R_2 + \ldots$

• Parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$

Short Circuits and Grounding

• Short circuit: Direct connection with little resistance, causing high current.

• Grounding: Provides a reference potential and safety.

RC Circuits

• Circuits with resistors and capacitors; charging and discharging described by exponential functions.

• Charging: $q(t) = Q(1 - e^{-t/RC})$

• Discharging: $q(t) = Q e^{-t/RC}$

Ch. 29: Magnetic Field

Magnetic Fields and Forces

Magnetic fields are produced by moving charges and currents. They exert forces on other moving charges and magnetic materials.

• Magnetic Field ($\vec{B}$): Vector field with units of tesla (T).

• Right-Hand Rule: Used to determine direction of $\vec{B}$ around a current-carrying wire.

• Force on a Moving Charge: $\vec{F} = q\vec{v} \times \vec{B}$

• Cyclotron Motion: Charged particles move in circles in a uniform $\vec{B}$ field.

• Radius of Path: $r = \frac{mv}{qB}$

• Angular Frequency: $\omega = \frac{qB}{m}$

Magnetic Field Due to a Current

• Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$

• Field of a Long Straight Wire: $B = \frac{\mu_0 I}{2\pi r}$

• Field of a Loop: $B = \frac{\mu_0 I}{2R}$ (at center)

Ampère's Law

• Relates the integrated magnetic field around a closed loop to the current passing through the loop: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$

Magnetic Properties of Matter

• Materials can be diamagnetic, paramagnetic, or ferromagnetic depending on their response to external magnetic fields.

• Ferromagnetic materials (e.g., iron) can retain permanent magnetization.

Ch. 30: Electromagnetic Induction

Faraday's Law of Induction

A changing magnetic field induces an electromotive force (emf) and current in a loop.

• Faraday's Law: $\mathcal{E} = -\frac{d\Phi_B}{dt}$, where $\Phi_B$ is magnetic flux.

• Lenz's Law: The induced emf opposes the change in magnetic flux.

Induced Electric Fields

• A changing magnetic field creates a circulating electric field, even in the absence of charges.

• $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$

Inductance

• Inductance ($L$): Measures the ability of a coil to induce emf due to changing current.

• $\mathcal{E}_L = -L \frac{dI}{dt}$

• Unit: Henry (H)

Energy Stored in Inductors

• $U = \frac{1}{2} L I^2$

LC Circuits

• Consist of an inductor and capacitor; oscillate at a natural frequency.

• Charge on capacitor: $Q(t) = Q_0 \cos(\omega t)$

• Current: $I(t) = -\omega Q_0 \sin(\omega t)$

• Frequency: $\omega = \frac{1}{\sqrt{LC}}$

Electromagnetic Waves

• Changing electric and magnetic fields propagate as waves at the speed of light: $c = 3 \times 10^8$ m/s.

Additional info: These notes cover the main concepts, definitions, and equations for current, resistance, DC circuits, magnetic fields, and electromagnetic induction, suitable for college-level physics exam preparation.