PHY2049 Formula Sheet and Study Guide: Chapters 21–36 (Electricity & Magnetism)

Useful Equations and Numbers

Fundamental Constants

Physics relies on several fundamental constants for calculations in electricity and magnetism. These constants are essential for solving problems and understanding physical laws.

• Coulomb's constant: $k = \frac{1}{4\pi\epsilon_0} = 8.988 \times 10^9 \text{ N} \cdot \text{m}^2 / \text{C}^2$

• Permittivity of free space: $\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2 / \text{N} \cdot \text{m}^2$

• Permeability of free space: $\mu_0 = 4\pi \times 10^{-7} \text{ T} \cdot \text{m} / \text{A}$

• Elementary charge: $e = 1.6 \times 10^{-19} \text{ C}$

• Electron mass: $m_e = 9.11 \times 10^{-31} \text{ kg}$

• Proton mass: $m_p = 1.67 \times 10^{-27} \text{ kg}$

• Speed of light: $c = 3.00 \times 10^8 \text{ m/s}$

• Acceleration due to gravity: $g = 9.8 \text{ m/s}^2$

Mathematics

Vectors

Vectors are quantities with both magnitude and direction, essential in physics for representing forces, fields, and motion.

• Vector notation: $\vec{A}, \vec{i}, \vec{j}, \vec{k}$

• Vector addition: $\vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k}$

• Scalar (dot) product: $\vec{A} \cdot \vec{B} = AB \cos \theta$

• Vector (cross) product: $\vec{A} \times \vec{B} = AB \sin \theta$

• Unit vectors: $\hat{i}, \hat{j}, \hat{k}$

Trigonometric Relationships

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Pythagorean Theorem

• $a^2 + b^2 = c^2$

Binomial Expansion

• $(a+b)^2 = a^2 + 2ab + b^2$

Power Rule

• $\frac{d}{dx} x^n = nx^{n-1}$

Geometry Formulas

• Circumference of a circle: $C = 2\pi r$

• Area of a circle: $A = \pi r^2$

• Surface area of a sphere: $A = 4\pi r^2$

• Surface area of a cylinder: $A = 2\pi r l$

• Volume of a sphere: $V = \frac{4}{3} \pi r^3$

Newton's Laws

• First Law: $\sum \vec{F} = 0$ (Object in equilibrium)

• Second Law: $\sum \vec{F} = m\vec{a}$ (Non-equilibrium)

Kinematic Equations

• $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• $a = \frac{v^2}{r}$ (circular motion)

• $v = v_0 + at$

Unit Prefixes

• $m = 10^{-3}$

• $cm = 10^{-2}$

• $\mu = 10^{-6}$

• $n = 10^{-9}$

• $p = 10^{-12}$

Units Table

The following table summarizes key physical quantities, their SI units, and equivalent unit names.

Symbol	Quantity	SI Unit	Unit Name / Equivalent
q	Charge	C	coulomb
r	Distance	m	meter
F	Force	N	newton
E	Electric field	N/C = V/m	newton per coulomb = volt per meter
λ	Linear charge density	C/m	coulomb per meter
σ	Surface charge density	C/m²	coulomb per square meter
ρ	Volume charge density	C/m³	coulomb per cubic meter
p	Electric dipole moment	C·m	coulomb–meter
τ	Torque	N·m	newton–meter
U	Energy / potential energy	J	joule
ΦE	Electric flux	N·m²/C	newton–meter² per coulomb
V	Electric potential	J/C = V	joule per coulomb = volt
I	Electric current	A	ampere
R	Resistance	Ω	ohm
C	Capacitance	F	farad
ρ	Resistivity	Ω·m	ohm–meter
A	Area	m²	square meter
B	Magnetic field	T = N·s/(C·m)	tesla = newton·second per coulomb·meter
ΦB	Magnetic flux	Wb = T·m²	weber = tesla·meter²
M	Inductance	H = Wb/A	henry = weber per ampere
L	Self-inductance	H = Wb/A	henry = weber per ampere
(I·SA)	Intensity of the wave	W/m²	watts per square meter

Chapter 21: Electric Forces and Fields

Coulomb's Law

Coulomb's law describes the force between two point charges.

• $F = k \frac{|q_1 q_2|}{r^2}$

• For vector sum of forces: $\vec{F} = \vec{F}_1 + \vec{F}_2 = F_1 \cos \theta + F_2 \cos \theta$

Electric Field

• For a point charge: $E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$

• Vector form: $\vec{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r}$

Superposition of Electric Fields

• Linear charge density: $\lambda = \frac{\text{charge}}{\text{length}}$

• Surface charge density: $\sigma = \frac{\text{charge}}{\text{area}}$

• Volume charge density: $\rho = \frac{\text{charge}}{\text{volume}}$

Electric Dipole Moment

• $p = qd$

• Direction: from negative to positive charge

Chapter 22: Gauss's Law and Electric Flux

Electric Flux

• Uniform field: $\Phi_E = E A \cos \theta$

• Nonuniform field: $\Phi_E = \int \vec{E} \cdot d\vec{A}$

Gauss's Law

• $\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{encl}}}{\epsilon_0}$

Electric Field for Various Charge Distributions

Charge Distribution	Point in Electric Field	Electric Field Magnitude
Single point charge q	Distance r from q	$E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$
Charge on surface of conducting sphere	Outside sphere, $r > R$	$E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$
Inside sphere, $r < R$		$E = 0$
Infinite wire	Distance r from wire	$E = \frac{\lambda}{2\pi\epsilon_0 r}$
Infinite conducting cylinder	Outside cylinder, $r > R$	$E = \frac{\lambda}{2\pi\epsilon_0 r}$
	Inside cylinder, $r < R$	$E = 0$
Solid insulating sphere	Outside sphere, $r > R$	$E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$
	Inside sphere, $r < R$	$E = \frac{1}{4\pi\epsilon_0} \frac{q r}{R^3}$
Infinite sheet of charge	Any point	$E = \frac{\sigma}{2\epsilon_0}$
Two oppositely charged plates	Any point between plates	$E = \frac{\sigma}{\epsilon_0}$
Charged conductor	Just outside the conductor	$E = \frac{\sigma}{\epsilon_0}$

Chapter 23: Electric Potential and Energy

Electric Potential Energy

• Between two points: $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r}$

• Collection of point charges: $U = \frac{1}{4\pi\epsilon_0} \sum_{i < j} \frac{q_i q_j}{r_{ij}}$

Work Done by a Conservative Force

• $W_{a \to b} = -\Delta U = U_a - U_b$

Electric Potential

• $V = \frac{U}{q_0}$

• For a point charge: $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$

Finding Electric Field from Potential

• $\vec{E} = -\nabla V$

• In Cartesian coordinates: $E_x = -\frac{\partial V}{\partial x}$, $E_y = -\frac{\partial V}{\partial y}$

Chapter 24: Capacitance

Capacitance

• $C = \frac{Q}{V}$

Parallel-Plate Capacitor

• $C = \epsilon_0 \frac{A}{d}$

Capacitors in Series and Parallel

• Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots$

• Parallel: $C_{eq} = C_1 + C_2 + \cdots$

Energy Stored in a Capacitor

• $U = \frac{Q^2}{2C} = \frac{1}{2} C V^2 = \frac{1}{2} Q V$

Energy Density in an Electric Field

• $u = \frac{1}{2} \epsilon_0 E^2$

Dielectrics

• Capacitance with dielectric: $C = K C_0 = K \epsilon_0 \frac{A}{d}$

• Dielectric constant: $K = \frac{C}{C_0} = \frac{V_0}{V}$

• Energy density: $u = \frac{1}{2} K \epsilon_0 E^2$

Chapter 25: Electric Current and Resistance

Electric Current

• $I = \frac{dQ}{dt}$

• Current density: $J = n q v_d$

Resistivity

• $\rho = \frac{E}{J}$

Temperature Dependence of Resistivity

Material	$\alpha$ ($^\circ$C$^{-1}$)	Material	$\alpha$ ($^\circ$C$^{-1}$)
Aluminum	0.0039	Lead	0.0030
Brass	0.0030	Manganin	0.000002
Carbon (graphite)	-0.0005	Mercury	0.0009
Constantan	0.00001	Nickrome	0.0004
Copper	0.0039	Silver	0.0038
Iron	0.0050	Tungsten	0.0045

Potential Difference for a Resistor

• $V = IR$

Resistance for a Cylindrical Conductor

• $R = \rho \frac{l}{A}$

Terminal Potential Difference

• $V_{ab} = E - Ir$

Current in a Circuit

• $E - Ir = IR$ or $I = \frac{E}{R + r}$

Power in Electric Circuits

• General element: $P = V_{ab} I = E I - I^2 r$

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

Chapter 26: DC Circuits

Resistors in Series and Parallel

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

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

Kirchhoff's Rules

• Junction rule: $\sum I = 0$

• Loop rule: $\sum V = 0$

RC Circuits

• Charging: $q = C \epsilon (1 - e^{-t/RC})$

• Discharging: $q = Q_0 e^{-t/RC}$

• Rate: $\frac{dq}{dt} = \frac{\epsilon}{R} e^{-t/RC}$

Chapter 27: Magnetic Forces and Fields

Magnetic Force on a Moving Charged Particle

• $\vec{F} = q \vec{v} \times \vec{B}$

• Magnitude: $F = q v B \sin \theta$

Magnetic Flux

• $\Phi_B = \int \vec{B} \cdot d\vec{A}$

Gauss's Law for Magnetism

• $\oint \vec{B} \cdot d\vec{A} = 0$

Circular Motion in a Uniform Magnetic Field

• $F = q v B$

• Radius: $r = \frac{m v}{q B}$

• Period: $T = \frac{2 \pi m}{q B}$

Magnetic Force in a Conductor

• $F = I l B \sin \theta$

Magnetic Torque on a Current Loop

• $\tau = N I A B \sin \theta$

Chapter 28: Sources of Magnetic Field

Biot–Savart Law

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

Magnetic Field of a Long, Straight, Current-Carrying Conductor

• $B = \frac{\mu_0 I}{2\pi r}$

Magnetic Field of a Current Loop

• $B = \frac{\mu_0 I}{2R}$ at the center

Ampère's Law

• $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{encl}}$

Magnetic Field of Solenoids

• $B = \mu_0 n I$

Chapter 29: Electromagnetic Induction

Faraday's Law

• $E = -N \frac{d\Phi_B}{dt}$

Motional EMF

• $E = B l v$

Maxwell's Equations

• $\nabla \cdot \vec{E} = \frac{Q_{\text{encl}}}{\epsilon_0}$ (Gauss's law for E fields)

• $\nabla \cdot \vec{B} = 0$ (Gauss's law for B fields)

• $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$ (Faraday's law)

• $\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$ (Ampère's law)

Chapter 30: Inductance

Mutual Inductance

• $\Phi_{B_2} = M_{21} I_1$

• $\epsilon_2 = -M_{21} \frac{dI_1}{dt}$

Self-Inductance

• $\Phi_B = L I$

• $\epsilon = -L \frac{dI}{dt}$

Additional info: This guide covers the essential formulas and concepts for electricity and magnetism, suitable for college-level introductory physics (PHY2049). It is organized by topic and chapter, with definitions, equations, and tables for quick reference and study.