MASTER — FAD1022 Complete Formula Sheet

The single most comprehensive formula sheet for the entire FAD1022 — Basic Physics II course. Compiled from all lectures (L1–L45), all tutorials (T4, T10, T12), and all concept sheets.

Universal Physical Constants

Symbol	Value	Description
$k$	$8.99 \times 10^9\ \text{N m}^2\text{ C}^{-2}$	Coulomb constant ($k = \frac{1}{4\pi\varepsilon_0}$)
$\varepsilon_0$	$8.85 \times 10^{-12}\ \text{F m}^{-1}$	Permittivity of free space
$\mu_0$	$4\pi \times 10^{-7}\ \text{H m}^{-1}$	Permeability of free space
$e$	$1.602 \times 10^{-19}\ \text{C}$	Elementary charge
$m_e$	$9.11 \times 10^{-31}\ \text{kg}$	Electron mass
$m_p$	$1.673 \times 10^{-27}\ \text{kg}$	Proton mass
$m_n$	$1.675 \times 10^{-27}\ \text{kg}$	Neutron mass
$h$	$6.626 \times 10^{-34}\ \text{J s}$	Planck constant
$\hbar$	$1.055 \times 10^{-34}\ \text{J s}$	Reduced Planck constant ($\hbar = h/2\pi$)
$c$	$3.00 \times 10^8\ \text{m s}^{-1}$	Speed of light in vacuum
$k_B$	$1.381 \times 10^{-23}\ \text{J K}^{-1}$	Boltzmann constant
$\sigma$	$5.67 \times 10^{-8}\ \text{W m}^{-2}\text{ K}^{-4}$	Stefan-Boltzmann constant
$b$	$2.90 \times 10^{-3}\ \text{m K}$	Wien's displacement constant
$N_A$	$6.022 \times 10^{23}\ \text{mol}^{-1}$	Avogadro's number
$1\ \text{eV}$	$1.602 \times 10^{-19}\ \text{J}$	Electron-volt
$1\ \text{u}$	$1.6606 \times 10^{-27}\ \text{kg}$	Atomic mass unit
$c^2$	$931.5\ \text{MeV/u}$	Energy equivalent of 1 u
$a_0$	$5.29 \times 10^{-11}\ \text{m}$	Bohr radius
$R_H$	$1.097 \times 10^7\ \text{m}^{-1}$	Rydberg constant
$\lambda_C$	$2.43 \times 10^{-12}\ \text{m}$	Compton wavelength of electron
$R_0$	$1.2 \times 10^{-15}\ \text{m} = 1.2\ \text{fm}$	Nuclear radius constant
$g$	$9.81\ \text{m s}^{-2}$	Acceleration due to gravity

1. Electrostatics

Charge Quantization

$$Q = ne \quad (n \in \mathbb{Z})$$

Coulomb's Law

$$F = \frac{1}{4\pi\varepsilon_0}\frac{Qq}{r^2} = \frac{kQq}{r^2}$$

Variable	Meaning	Unit
$F$	Electrostatic force magnitude	N
$Q, q$	Point charge magnitudes	C
$r$	Separation distance	m

Like charges repel; opposite charges attract.

Electric Field Strength

Definition: $$E = \frac{F}{q_0}$$

Due to a point charge: $$E = \frac{kQ}{r^2} = \frac{Q}{4\pi\varepsilon_0 r^2}$$

Variable	Meaning	Unit
$E$	Electric field strength	N C$^{-1}$ or V m$^{-1}$
$q_0$	Test charge	C
$Q$	Source charge	C
$r$	Distance from source charge	m

Positive charge: $\vec{E}$ points radially outward.
Negative charge: $\vec{E}$ points radially inward.

Principle of Superposition

$$\vec{E}_{\text{net}} = \sum_i \vec{E}_i$$

$$E_{\text{net},x} = \sum E_{i,x}, \quad E_{\text{net},y} = \sum E_{i,y}$$

$$E_{\text{net}} = \sqrt{E_{\text{net},x}^2 + E_{\text{net},y}^2}, \quad \theta = \tan^{-1}!\left(\frac{E_{\text{net},y}}{E_{\text{net},x}}\right)$$

Neutral Point

$$\vec{E}_{\text{net}} = 0$$

Occurs between two like charges, closer to the smaller charge.

Electric Force on a Charge in a Field

$$\vec{F} = q\vec{E}$$

Particle	Charge	Force Magnitude	Direction
Electron	$-e$	$F = Ee$	Opposite to $\vec{E}$
Proton	$+e$	$F = Ee$	Same as $\vec{E}$
Alpha ($^4_2\text{He}$)	$+2e$	$F = E(2e)$	Same as $\vec{E}$

Charge Motion in Uniform Electric Field

Dynamic Equilibrium (Horizontal)

$$qE = mg$$

Perpendicular Entry (Parabolic Trajectory)

$$a_y = \frac{q_0 E}{m}, \quad v_x = v_0, \quad t = \frac{x}{v_0}$$

$$v_y = \frac{q_0 E x}{m v_0}, \quad v = \sqrt{v_x^2 + v_y^2}, \quad \theta = \tan^{-1}!\left(\frac{v_y}{v_x}\right)$$

$$s_y = -\frac{1}{2} a_y t^2$$

Parallel Entry (Linear Acceleration)

$$a = \frac{qE}{m}$$

Kinematic Equations (SUVAT)

$$v = u + at, \quad s = ut + \frac{1}{2}at^2, \quad v^2 = u^2 + 2as$$

Electric Field Line Properties

$\vec{E}$ is tangent to field lines.
$|\vec{E}| \propto$ line density (closer = stronger).
Lines start on $+$, end on $-$.
Number of lines $\propto$ charge magnitude.
Field lines never cross.

2. Capacitors & Dielectrics

Capacitance Definition

$$C = \frac{Q}{\Delta V}$$

Symbol	Meaning	Unit
$C$	Capacitance	F (Farad)
$Q$	Charge on one plate	C
$\Delta V$	Potential difference	V

Parallel-Plate Capacitor (Vacuum/Air)

Electric field: $$E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{A\varepsilon_0} = \frac{\Delta V}{d}$$

Capacitance: $$C = \frac{A\varepsilon_0}{d}$$

Symbol	Meaning	Unit
$A$	Plate area	m$^2$
$d$	Plate separation	m
$\sigma$	Surface charge density ($Q/A$)	C m$^{-2}$

Capacitors with Dielectrics

Dielectric constant: $$\kappa = \frac{\varepsilon}{\varepsilon_0}$$

Capacitance with dielectric: $$C = \kappa C_0 = \kappa \frac{A\varepsilon_0}{d}$$

Isolated Capacitor (Battery Disconnected)

Quantity	Relation
Charge	$Q = Q_0$ (constant)
Voltage	$\Delta V = \dfrac{\Delta V_0}{\kappa}$
Capacitance	$C = \kappa C_0$
Electric field	$\vec{E} = \dfrac{\vec{E}_0}{\kappa}$

Energy Stored in a Charged Capacitor

$$U = W = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}C(\Delta V)^2 = \frac{1}{2}Q,\Delta V$$

RC Circuit Time Constant

$$\tau = RC$$

Symbol	Meaning	Unit
$\tau$	Time constant	s
$R$	Resistance	$\Omega$
$C$	Capacitance	F

3. DC Circuits

Ohm's Law

$$V = IR, \quad I = \frac{V}{R}, \quad R = \frac{V}{I}$$

Electromotive Force (EMF) & Terminal Voltage

Terminal voltage: $$V_{\text{terminal}} = \varepsilon - Ir$$

Current from real source: $$I = \frac{\varepsilon}{R_{\text{external}} + r}$$

Alternative forms: $$V_{\text{terminal}} = IR_{\text{external}} = \varepsilon \frac{R_{\text{external}}}{R_{\text{external}} + r}$$

Symbol	Meaning	Unit
$\varepsilon$	EMF	V
$r$	Internal resistance	$\Omega$
$R_{\text{external}}$	External load resistance	$\Omega$

Power in DC Circuits

General: $$P = IV$$

Dissipated in resistor: $$P = I^2 R = \frac{V^2}{R}$$

Supplied by source: $$P_{\text{source}} = I\varepsilon$$

Lost to internal resistance: $$P_{\text{lost}} = I^2 r$$

Delivered to load: $$P_{\text{delivered}} = I^2 R_{\text{external}} = IV_{\text{terminal}}$$

Resistor Combinations

Series: $$R_{\text{eq}} = R_1 + R_2 + R_3 + \dots = \sum_i R_i$$

Parallel: $$\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots = \sum_i \frac{1}{R_i}$$

Two resistors in parallel: $$R_{\text{eq}} = \frac{R_1 R_2}{R_1 + R_2}$$

Current Divider (Two Parallel Resistors)

$$I_1 = I_{\text{total}} \frac{R_2}{R_1 + R_2}, \quad I_2 = I_{\text{total}} \frac{R_1}{R_1 + R_2}$$

Kirchhoff's Laws

KCL (Junction Rule): $$\sum I_{\text{in}} = \sum I_{\text{out}} \quad \text{or} \quad \sum I = 0$$

KVL (Loop Rule): $$\sum_{\text{loop}} \Delta V = 0$$

Single Loop Circuit

$$I = \frac{\sum \varepsilon}{\sum R}$$

4. AC Analysis (Basics, Phasors, Reactance)

AC Signal Fundamentals

Instantaneous current & voltage: $$I(t) = I_0 \sin(\omega t), \quad V(t) = V_0 \sin(\omega t)$$

Angular frequency: $$\omega = \frac{2\pi}{T} = 2\pi f$$

Symbol	Meaning	Unit
$I_0, V_0$	Peak (maximum) values	A, V
$\omega$	Angular frequency	rad/s
$f$	Frequency	Hz
$T$	Period	s

RMS Values

$$I_{\text{rms}} = \frac{I_{\text{max}}}{\sqrt{2}} = 0.707, I_{\text{max}}$$

$$V_{\text{rms}} = \frac{V_{\text{max}}}{\sqrt{2}} = 0.707, V_{\text{max}}$$

Power in AC Circuits

$$P = V_{\text{rms}}, I_{\text{rms}}$$

$$I_{\text{rms}} = \frac{P}{V_{\text{rms}}}, \quad I_{\text{max}} = I_{\text{rms}} \times \sqrt{2}$$

Phasors & Phase Angle

General sinusoidal form: $$A(t) = A_m \sin(\omega t + \phi)$$

Leading (positive $\phi$): $A(t) = A_m \sin(\omega t + \phi)$

Lagging (negative $\phi$): $A(t) = A_m \sin(\omega t - \phi)$

Impedance

$$Z = \frac{V_{\text{rms}}}{I_{\text{rms}}} = \frac{V_0}{I_0}$$

Pure Resistive Circuit (PRC)

$$I = I_0 \sin(\omega t), \quad V_R = V_0 \sin(\omega t)$$

Phase difference: $\Delta\phi = 0$ (in phase)
Impedance: $Z = R$

Pure Capacitive Circuit (PCC)

$$V_C = V_0 \sin(\omega t), \quad I = I_0 \sin\left(\omega t + \frac{\pi}{2}\right)$$

Current leads voltage by $90°$ ($\pi/2$ rad)
Capacitive reactance: $$X_C = \frac{1}{2\pi f C} = \frac{1}{\omega C} = \frac{V_{\text{rms}}}{I_{\text{rms}}}$$
$X_C \propto \dfrac{1}{f}$

Pure Inductive Circuit (PLC)

$$V = V_0 \sin\left(\omega t + \frac{\pi}{2}\right), \quad I = I_0 \sin(\omega t)$$

Voltage leads current by $90°$ ($\pi/2$ rad)
Inductive reactance: $$X_L = 2\pi f L = \omega L = \frac{V_{\text{rms}}}{I_{\text{rms}}}$$
$X_L \propto f$

Memory Aid — CIVIL Mnemonic

Capacitor: I leads V
Inductor: V leads I
Resistor: In phase

5. AC Series Circuits (RL, RC, RLC, Resonance, Power)

Reactance Summary

$$X_L = \omega L = 2\pi f L$$

$$X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$$

$$X = X_L - X_C \quad \text{(net reactance)}$$

RL Series Circuit

$$I_T = I_R = I_L$$

$$V_T = \sqrt{V_R^2 + V_L^2}$$

$$V_R = I_{\text{rms}} R, \quad V_L = I_{\text{rms}} X_L$$

$$Z = \sqrt{R^2 + X_L^2}$$

$$\theta = \tan^{-1}!\left(\frac{V_L}{V_R}\right) = \tan^{-1}!\left(\frac{X_L}{R}\right) \quad (>0)$$

$$I_{\text{rms}} = \frac{V_{\text{rms}}}{Z}$$

Voltage leads current by $\theta$.

RC Series Circuit

$$I_T = I_R = I_C$$

$$V_T = \sqrt{V_R^2 + V_C^2}$$

$$V_R = I_{\text{rms}} R, \quad V_C = I_{\text{rms}} X_C$$

$$Z = \sqrt{R^2 + X_C^2}$$

$$\theta = \tan^{-1}!\left(\frac{-X_C}{R}\right) = \tan^{-1}!\left(\frac{V_C}{V_R}\right) \quad (<0)$$

$$I_{\text{rms}} = \frac{V_{\text{rms}}}{Z}$$

Current leads voltage by $|\theta|$.
Finding $C$ from phase angle: $$\tan|\theta| = \frac{X_C}{R} \Rightarrow X_C = R\tan|\theta| \Rightarrow C = \frac{1}{2\pi f X_C}$$

RLC Series Circuit

$$I_T = I_R = I_L = I_C$$

$$V_T = \sqrt{V_R^2 + (V_L - V_C)^2}$$

$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$

$$\theta = \tan^{-1}!\left(\frac{X_L - X_C}{R}\right)$$

$\theta > 0$: inductive (voltage leads)
$\theta < 0$: capacitive (current leads)
$\theta = 0$: resistive / resonance

$$V_R = I_{\text{rms}} R, \quad V_L = I_{\text{rms}} X_L, \quad V_C = I_{\text{rms}} X_C$$

$$I_{\text{rms}} = \frac{V_{\text{rms}}}{Z}$$

RLC Series Resonance

Resonance condition: $$X_L = X_C$$

Resonant frequency: $$f_0 = \frac{1}{2\pi\sqrt{LC}}$$

Capacitance at resonance: $$C = \frac{1}{4\pi^2 f_0^2 L}$$

Properties at resonance:

Property	Formula
Impedance (minimum)	$Z = R$
Current (maximum)	$I_{\text{rms}} = \dfrac{V_{\text{rms}}}{R}$
Phase angle	$\theta = 0°$
Power factor	$\text{PF} = 1$ (unity)
Circuit behavior	Purely resistive

Reactance ratio at different frequencies: $$\frac{X_C}{X_L} = \left(\frac{f_0}{f}\right)^2$$

AC Power & Power Factor

Three Types of Power

Type	Symbol	Unit	Dissipated By
Average (Real) Power	$P_{\text{ave}}$	W	Resistor only
Reactive Power	$P_R$ or $Q$	VAr	Inductor / Capacitor
Apparent Power	$P_A$ or $S$	VA	Impedance $Z$

Power Factor

$$\text{PF} = \cos\phi = \frac{P_{\text{ave}}}{P_A} = \frac{R}{Z}$$

Power Triangle

$$P_A^2 = P_{\text{ave}}^2 + P_R^2$$

$$P_A = \sqrt{P_{\text{ave}}^2 + P_R^2}$$

Power by Circuit Type

Pure R: $$P_{\text{ave}} = V_{\text{rms}} I_{\text{rms}}$$

Pure L / Pure C: $$P_{\text{ave}} = 0$$ $$P_r = I_{\text{rms}}^2 X_L \quad \text{or} \quad P_r = I_{\text{rms}}^2 X_C$$

RL / RC: $$P_{\text{ave}} = I_{\text{rms}}^2 R$$ $$P_R = I_{\text{rms}}^2 X_L \quad \text{or} \quad P_R = I_{\text{rms}}^2 X_C$$ $$P_A = I_{\text{rms}}^2 Z = \sqrt{P_{\text{ave}}^2 + P_R^2}$$

RLC: $$P_{\text{ave}} = I_{\text{rms}}^2 R$$ $$P_R = I_{\text{rms}}^2 |X_L - X_C|$$ $$P_A = I_{\text{rms}}^2 Z = \sqrt{P_{\text{ave}}^2 + P_R^2}$$

General Power Formulas

$$P_{\text{ave}} = V_{\text{rms}} I_{\text{rms}} \cos\phi$$ $$P_R = V_{\text{rms}} I_{\text{rms}} \sin\phi$$ $$P_A = V_{\text{rms}} I_{\text{rms}}$$

Phase Angle Summary

Circuit	Phase Angle	Sign	Leading Signal
Pure R	$0°$	—	In phase
Pure L	$+90°$	Positive	Voltage leads
Pure C	$-90°$	Negative	Current leads
RL Series	$\tan^{-1}(X_L/R)$	Positive	Voltage leads
RC Series	$\tan^{-1}(-X_C/R)$	Negative	Current leads
RLC Series	$\tan^{-1}\big((X_L-X_C)/R\big)$	$\pm$	Depends on $X_L$ vs $X_C$
RLC at Resonance	$0°$	—	In phase

Impedance Summary

Circuit	Impedance Formula
Pure R	$Z = R$
Pure L	$Z = X_L$
Pure C	$Z = X_C$
RL Series	$Z = \sqrt{R^2 + X_L^2}$
RC Series	$Z = \sqrt{R^2 + X_C^2}$
RLC Series	$Z = \sqrt{R^2 + (X_L - X_C)^2}$
RLC at Resonance	$Z = R$

6. Magnetism

Unit Conversions

$$1\ \text{T} = 1\ \frac{\text{N}}{\text{A}\cdot\text{m}}, \quad 1\ \text{G} = 10^{-4}\ \text{T}$$

Magnetic Field due to Current-Carrying Conductors

Long Straight Wire

$$B = \frac{\mu_0 I}{2\pi r}$$

$B \propto \dfrac{1}{r}$

Circular Loop (at Centre)

$$B = \frac{\mu_0 N I}{2r}$$

Long Solenoid (at Centre)

$$B = \frac{\mu_0 N I}{L} = \mu_0 n I$$

$n = N/L$ = turns per unit length
Valid when $L \gg r$
Field inside uniform; outside $\approx 0$

Superposition of Magnetic Fields

$$\vec{B}_{\text{net}} = \sum_i \vec{B}_i$$

Force on a Moving Charge (Lorentz Force)

Vector form: $$\vec{F}_B = q\vec{v} \times \vec{B}$$

Magnitude: $$|F_B| = |q|vB\sin\theta$$

Maximum: $\theta = 90° \Rightarrow |F_B| = |q|vB$
Zero: $\theta = 0°$ or $v = 0$

Motion of Charged Particle in Uniform $B$-Field

When $v \perp B$: $$qvB = \frac{mv^2}{r}$$

Radius: $$r = \frac{mv}{qB}$$

Velocity from radius: $$v = \frac{rqB}{m}$$

Period: $$T = \frac{2\pi r}{v} = \frac{2\pi m}{qB}$$

Period is independent of velocity.

Angular frequency (cyclotron frequency): $$\omega = \frac{qB}{m}$$

Velocity Selector

$$qE = qvB \Rightarrow v = \frac{E}{B}$$

Mass Spectrometer

$$m = \frac{qB'B^2r}{E}$$

If same field $B$ in both regions: $$m = \frac{qrB^2}{E}$$

Force Between Two Parallel Wires

Force magnitude: $$F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$$

Force per unit length: $$f = \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$

Same direction currents $\rightarrow$ attractive
Opposite direction currents $\rightarrow$ repulsive

Ampere's Law

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

Magnetic Field of Cylindrical Wire

Region	Formula
Inside ($r < R$)	$B = \dfrac{\mu_0 I r}{2\pi R^2}$
At surface ($r = R$)	$B = \dfrac{\mu_0 I}{2\pi R}$
Outside ($r > R$)	$B = \dfrac{\mu_0 I}{2\pi r}$

Enclosed Current Inside Wire

$$I_{\text{enc}} = \frac{r^2}{R^2} I$$

Torque on a Current Loop

$$\tau = N I A B \sin\theta$$

Maximum: $\theta = 90° \Rightarrow \tau_{\max} = N I A B$
Zero: $\theta = 0° \Rightarrow \tau = 0$

Direction Rules

Rule	Application
Right Hand Grip Rule	Field around wire (thumb = $I$, fingers = $\vec{B}$)
Right Hand Rule	Force on $+q$ (fingers = $\vec{v}$, curl to $\vec{B}$, thumb = $\vec{F}$)
Fleming's Left Hand Rule	First = $\vec{B}$, second = $\vec{v}$, thumb = $\vec{F}$
Negative charge	Reverse the $+q$ force direction

$\odot$ = out of page; $\otimes$ = into page

7. Electromagnetic Induction

Magnetic Flux

$$\Phi_B = B A \cos\theta$$

$$\Delta\Phi_B = \Phi_{B,f} - \Phi_{B,i} = B_f A_f \cos\theta_f - B_i A_i \cos\theta_i$$

Area of circular loop: $$A = \pi r^2$$

Faraday's Law of Induction

Instantaneous: $$\varepsilon = -N \frac{d\Phi_B}{dt}$$

Average: $$\varepsilon_{\text{avg}} = -N \frac{\Delta\Phi_B}{\Delta t}$$

Motional EMF

$$\varepsilon = B l v$$

Condition: $\vec{B}$, $\vec{l}$, $\vec{v}$ mutually perpendicular.

Induced Current

$$I = \frac{\varepsilon}{R}$$

Lenz's Law

The induced current flows in a direction that opposes the change in magnetic flux that produced it.

Power in Electromagnetic Systems

Dissipated: $$P = I^2 R = \frac{\varepsilon^2}{R}$$

Mechanical input: $$P = F v$$

Magnetic Force on Current-Carrying Conductor

$$F = I l B = \frac{B^2 l^2 v}{R}$$

Peak EMF in Rotating Coil

$$\varepsilon_{\max} = N B A \omega$$

$$\omega = \frac{\Delta\theta}{\Delta t}$$

8. Inductance & Transformers

Self-Inductance

Magnetic field of solenoid: $$B = \mu_0 n I = \mu_0 \left(\frac{N}{\ell}\right) I$$

Self-induced EMF (Back EMF): $$\mathcal{E} = -L \frac{dI}{dt}$$

Inductance of a solenoid: $$L = \frac{\mu_0 N^2 A}{\ell}$$

General definition: $$L = \frac{N\Phi_B}{I}$$

Energy Stored in an Inductor

$$U = \frac{1}{2} L I^2$$

Capacitor–Inductor Analogies

Property	Capacitor	Inductor
Geometry	$C = \dfrac{\varepsilon_0 A}{d}$	$L = \dfrac{\mu_0 N^2 A}{\ell}$
Energy	$U = \dfrac{1}{2} C V^2$	$U = \dfrac{1}{2} L I^2$
Definition	$C = \dfrac{Q}{V}$	$L = \dfrac{N\Phi}{I}$

Mutual Inductance

Definition: $$M_{21} = \frac{N_2 \Phi_{21}}{i_1}, \quad M_{12} = \frac{N_1 \Phi_{12}}{i_2}$$

Reciprocity: $$M_{12} = M_{21} = M$$

Induced EMF: $$\varepsilon_2 = M \frac{di_1}{dt}, \quad \varepsilon_1 = M \frac{di_2}{dt}$$

Mutual inductance for coaxial solenoids: $$M = \frac{\mu_0 N_p N_s A}{l}$$

Transformers

Fundamental principle: $$\frac{d\Phi_1}{dt} = \frac{d\Phi_2}{dt}$$

Faraday's law: $$V_s = -N_s \frac{d\Phi}{dt}, \quad V_p = -N_p \frac{d\Phi}{dt}$$

Voltage–turns ratio: $$\frac{V_s}{V_p} = \frac{N_s}{N_p}$$

Current ratio (ideal): $$\frac{I_s}{I_p} = \frac{V_p}{V_s} = \frac{N_p}{N_s}$$

Power conservation (ideal): $$P_p = P_s \implies V_p I_p = V_s I_s$$

Step-up: $N_s > N_p$ (voltage increases, current decreases)
Step-down: $N_s < N_p$ (voltage decreases, current increases)

Transformer Energy Losses (Real Transformers)

Copper loss: $$P_{\text{copper}} = I^2 R$$

Eddy current loss: $$P_{\text{eddy}} \propto B^2 f^2 t^2$$

Fix: laminated iron core

Hysteresis loss: Energy lost during magnetization reversal

Fix: soft magnetic materials (Silicon Steel)

Flux leakage: Not all flux reaches secondary coil

Fix: wrap coils on top of each other

Power Transmission

Power loss in lines: $$P_{\text{loss}} = I^2 R_{\text{line}}$$

Percentage power loss: $$% \text{ Power Loss} = \frac{I^2 R_{\text{line}}}{P_{\text{total}}} \times 100%$$

RMS voltage: $$V_{\text{rms}} = \frac{V_{\text{max}}}{\sqrt{2}}$$

9. Semiconductors & Op-Amps

Diodes

Forward Bias Voltage Drops

Semiconductor	$V_D$
Germanium (Ge)	$0.3\ \text{V}$
Silicon (Si)	$0.7\ \text{V}$
Gallium Arsenide (GaAs)	$1.5\ \text{V}$

Diode DC Series Configuration

Forward bias (ON): $$E - V_R - V_D = 0$$

Reverse bias (OFF): $$I_D = I_R = 0, \quad V_R = 0, \quad V_D = E$$

Half-Wave Rectifier

Positive cycle: $$V_m - V_D - V_O = 0 \Rightarrow V_O = V_m - V_D$$

Negative cycle: $$V_O = 0$$

Average DC output voltage: $$V_{DC} = 0.318,(V_m - V_D) = \frac{V_O}{\pi}$$

Full-Wave Rectifier Average DC Output

$$V_{DC} = \frac{2V_O}{\pi}$$

Ohm's Law for Diode Circuits

$$I = \frac{V_{\text{supply}} - V_D}{R}$$

$$V_R = IR$$

Bipolar Junction Transistor (BJT)

Fundamental Relationships

$$I_E = I_B + I_C$$

$$\beta = \frac{I_C}{I_B} \Rightarrow I_C = \beta I_B$$

$$\alpha = \frac{I_C}{I_E}$$

$$I_E = (\beta + 1) I_B$$

Operating Regions

Region	$V_{BE}$	$V_{CE}$	$I_C$	Application
Cutoff	$< 0.7\ \text{V}$	$= V_{CC}$	$\approx 0$	Open switch
Active	$\approx 0.7\ \text{V}$	$> V_{CE(\text{sat})}$	$\beta I_B$	Amplifier
Saturation	$\approx 0.7\ \text{V}$	$\approx 0.2\ \text{V}$	$I_{C(\text{sat})}$	Closed switch

Fixed-Bias Circuit

Base current: $$I_B = \frac{V_{CC} - V_{BE}}{R_B}$$

Collector-emitter voltage: $$V_{CE} = V_{CC} - I_C R_C$$

Saturation current: $$I_{C(\text{sat})} = \frac{V_{CC}}{R_C}$$

Emitter-Stabilized Bias

Base current: $$I_B = \frac{V_{CC} - V_{BE}}{R_B + (\beta + 1)R_E}$$

Collector-emitter voltage: $$V_{CE} = V_{CC} - I_C(R_C + R_E)$$

Saturation current: $$I_{C(\text{sat})} = \frac{V_{CC}}{R_C + R_E}$$

Node voltages: $$V_E = I_E R_E, \quad V_C = V_{CC} - I_C R_C, \quad V_B = V_{BE} + V_E$$

Stability condition: $$(\beta + 1)R_E \geq 10 R_B \Rightarrow I_C \approx \frac{V_{CC} - V_{BE}}{R_E}$$

Voltage Divider Bias (Approximate Analysis)

Validity condition: $$\beta R_E \geq 10 R_{B2} \quad \text{or} \quad R_{TH} \leq 0.1, \beta R_E$$

$$R_{TH} = R_{B1} \parallel R_{B2} = \frac{R_{B1} R_{B2}}{R_{B1} + R_{B2}}$$

Analysis steps: $$V_B = \frac{R_{B2}}{R_{B1} + R_{B2}} V_{CC}$$

$$V_E = V_B - V_{BE}$$

$$I_E = \frac{V_E}{R_E} \approx I_C$$

$$V_{CE} = V_{CC} - I_C(R_C + R_E)$$

Operational Amplifiers (Op-Amps)

Inverting Amplifier

$$V_{\text{out}} = -\frac{R_f}{R_1} V_{\text{in}}$$

$180°$ phase inversion

Non-Inverting Amplifier

$$V_{\text{out}} = \left(1 + \frac{R_f}{R_1}\right) V_{\text{in}} = \left(\frac{R_1 + R_f}{R_1}\right) V_{\text{in}}$$

Output in phase with input; gain $\geq 1$

Comparison

Configuration	Gain Formula	Phase
Inverting	$-\dfrac{R_f}{R_1}$	$180°$
Non-inverting	$1 + \dfrac{R_f}{R_1}$	$0°$

10. Atomic Physics

Bohr Model of the Hydrogen Atom

Angular Momentum Quantization

$$L = mvr = n\frac{h}{2\pi} = n\hbar$$

Coulomb Force as Centripetal Force

$$\frac{e^2}{4\pi\varepsilon_0 r^2} = \frac{mv^2}{r}$$

Bohr Orbital Radius

$$r_n = \left(\frac{\varepsilon_0 h^2}{\pi m e^2}\right) n^2 = a_0 n^2 = (5.29 \times 10^{-11}\ \text{m}), n^2$$

Total Energy

$$E = K + U = \frac{1}{2}mv^2 - \frac{ke^2}{r} = -\frac{ke^2}{2r}$$

Quantized Energy Levels

$$E_n = -\left(\frac{2\pi^2 m k^2 e^4}{h^2}\right)\frac{1}{n^2} = -(13.6\ \text{eV})\frac{1}{n^2}$$

Key values:

$n$	State	$E_n$
1	Ground	$-13.6\ \text{eV}$
2	1st excited	$-3.4\ \text{eV}$
3	2nd excited	$-1.51\ \text{eV}$
4	3rd excited	$-0.85\ \text{eV}$

Ionization Energy

$$\Delta E = E_\infty - E_1 = 13.6\ \text{eV}$$

Atomic Transitions & Photon Energy

$$\Delta E = hf = E_i - E_f$$

$$E_{\text{light}} = -13.6\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)\ \text{eV}$$

Rydberg Formula

$$\frac{1}{\lambda} = R_H\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$$

Spectral series:

Lyman: $n_f = 1$ (UV)
Balmer: $n_f = 2$ (visible)
Paschen: $n_f = 3$ (IR)

Quantum Numbers

Symbol	Name	Allowed Values
$n$	Principal	$1, 2, 3, \dots$
$l$	Orbital (azimuthal)	$0, 1, 2, \dots, (n-1)$
$m_l$	Magnetic	$-l, \dots, 0, \dots, +l$
$m_s$	Spin	$+\frac{1}{2}, -\frac{1}{2}$

Radiation Processes

Stimulated absorption: Electron absorbs photon and moves to higher state. $$E_{\text{photon}} = E_{\text{higher}} - E_{\text{lower}}$$

Spontaneous emission: Excited electron emits photon randomly. $$E_{\text{photon}} = E_{\text{excited}} - E_{\text{ground}}$$

Stimulated emission: Incident photon triggers identical photon emission.

LASER requires population inversion + optical feedback.

Reduced Mass (Two-Body System)

$$\mu = \frac{m_e M}{m_e + M}$$

11. Nuclear Physics

Nuclear Structure

Nuclide Notation

$$^A_Z X, \quad N = A - Z$$

Symbol	Description
$A$	Mass number (protons + neutrons)
$Z$	Atomic number (protons)
$N$	Neutron number

Nuclear Radius

$$R = R_0 A^{1/3}$$

Nuclear Volume

$$V = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi A R_0^3$$

Nuclear Density

$$\rho \approx 2.3 \times 10^{17}\ \text{kg/m}^3$$

Mass-Energy Equivalence

Einstein's Relation

$$E = mc^2$$

Atomic Mass Unit

$$1\ \text{u} = 1.6606 \times 10^{-27}\ \text{kg}$$

$$1\ \text{u} = 931.5\ \text{MeV}/c^2$$

Energy Conversions

$$1\ \text{eV} = 1.602 \times 10^{-19}\ \text{J}$$

$$1\ \text{MeV} = 10^6\ \text{eV} = 1.602 \times 10^{-13}\ \text{J}$$

Mass Defect & Binding Energy

Mass Defect

$$\Delta m = Zm_p + Nm_n - m_N$$

Binding Energy

$$E_B = (\Delta m)c^2 = [Zm_p + Nm_n - m_N]c^2$$

In MeV: $$E_B = \Delta m \times 931.5\ \text{MeV/u}$$

Binding Energy per Nucleon

$$\frac{E_B}{A}$$

Peak stability at Fe-56: $\approx 8.8\ \text{MeV/nucleon}$

Radioactive Decay

Decay Modes

Mode	Particle	$\Delta Z$	$\Delta A$	Condition
Alpha ($\alpha$)	$^4_2\text{He}$	$-2$	$-4$	Nucleus too heavy
Beta minus ($\beta^-$)	$^0_{-1}e$	$+1$	$0$	Too many neutrons
Positron ($\beta^+$)	$^0_{+1}e$	$-1$	$0$	Too many protons
Gamma ($\gamma$)	Photon	$0$	$0$	Excited nucleus

Decay Law

$$\frac{dN}{dt} = -\lambda N$$

$$N(t) = N_0 e^{-\lambda t}$$

Activity

$$A = \lambda N = -\frac{dN}{dt}$$

$$A_0 = \lambda N_0$$

$$A = A_0 e^{-\lambda t}$$

Activity Units

$$1\ \text{Bq} = 1\ \text{decay s}^{-1}$$

$$1\ \text{Ci} = 3.70 \times 10^{10}\ \text{Bq}$$

Half-Life

$$T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$$

$$\lambda = \frac{0.693}{T_{1/2}}$$

Fraction Remaining

$$\frac{N(t)}{N_0} = e^{-\lambda t} = \left(\frac{1}{2}\right)^{t/T_{1/2}}$$

Carbon Dating

$$\lambda = \frac{\ln 2}{5730\ \text{yr}} = 1.21 \times 10^{-4}\ \text{yr}^{-1}$$

$$t = -\frac{1}{\lambda}\ln\left(\frac{N(t)}{N_0}\right) = \frac{1}{\lambda}\ln\left(\frac{N_0}{N(t)}\right)$$

$$t = \frac{T_{1/2} \cdot \ln(N_0/N)}{\ln 2}$$

Number of Nuclei in a Sample

$$N_0 = \frac{N_A}{M} \times m_{\text{sample}}$$

Nuclear Reactions & Q-Value

Conservation Laws

Conservation of charge ($Z$)
Conservation of mass number ($A$)
Conservation of energy

Q-Value

$$\Delta m = \sum m_{\text{before}} - \sum m_{\text{after}}$$

$$Q = (\Delta m)c^2 = \Delta m \times 931.5\ \text{MeV}$$

$Q > 0$: Exothermic (energy released)
$Q < 0$: Endothermic (energy absorbed)

Nuclear Fusion

Definition: Small nuclei combine to form larger nuclei, releasing energy.

Energy released: $$\text{Energy} = (\text{Total } E_B \text{ of products}) - (\text{Total } E_B \text{ of reactants})$$

Or via mass defect: $$\Delta m = (\text{total mass of reactants}) - (\text{total mass of products})$$

$$E = \Delta m , c^2 = \Delta m \times 931.5\ \text{MeV}$$

Nuclear Fission

Definition: Heavy nucleus splits into two lighter nuclei.

Energy calculation: $$\Delta m = (m_{\text{parent}} + m_n) - (m_{\text{product 1}} + m_{\text{product 2}} + \text{neutrons})$$

$$Q = \Delta m \times 931.5\ \text{MeV}$$

12. Modern Physics & Quantum Mechanics

Black Body Radiation

Energy Conservation for Incident Radiation

$$\alpha_\nu + \rho_\nu + \tau_\nu = 1$$

$\alpha_\nu$ = absorptivity, $\rho_\nu$ = reflectivity, $\tau_\nu$ = transmissivity
Perfect black body: $\alpha_\nu = 1$

Planck's Quantum Hypothesis

$$E = hf = \hbar\omega = \frac{hc}{\lambda}$$

Wien's Displacement Law

$$\lambda_{\text{max}} = \frac{b}{T}$$

As $T$ increases, peak wavelength shifts to shorter wavelengths.

Stefan-Boltzmann Law

$$\frac{P}{A} = \sigma T^4$$

Total radiated power: $$P = \sigma A T^4$$

Planck's Law (Spectral Energy Density)

$$u(\lambda, T) = \frac{8\pi hc}{\lambda^5}\frac{1}{e^{hc/\lambda k_B T} - 1}$$

Photons & Photoelectric Effect

Photon Energy

$$E = hf = \frac{hc}{\lambda}$$

Frequency-Wavelength Relationship

$$c = f\lambda$$

Work Function

$$\phi = hf_0 = \frac{hc}{\lambda_c}$$

Threshold Frequency

$$f_0 = \frac{\phi}{h}$$

Cutoff Wavelength

$$\lambda_c = \frac{hc}{\phi}$$

Einstein's Photoelectric Equation

$$KE_{\max} = hf - \phi = \frac{hc}{\lambda} - \phi$$

Maximum Kinetic Energy from Velocity

$$KE_{\max} = \frac{1}{2} m_e v_{\max}^2$$

Stopping Potential

$$KE_{\max} = eV_s$$

$$V_s = \frac{hf - \phi}{e} = \frac{KE_{\max}}{e}$$

Emission Conditions

Condition	Result
$hf < \phi$ or $f < f_0$ or $\lambda > \lambda_c$	No electrons emitted
$hf = \phi$ or $f = f_0$ or $\lambda = \lambda_c$	Emission with $KE_{\max} = 0$
$hf > \phi$ or $f > f_0$ or $\lambda < \lambda_c$	Electrons emitted with $KE_{\max} = hf - \phi$

Frequency controls whether electrons are emitted.
Intensity controls how many electrons are emitted.

Compton Effect

Compton Wavelength Shift

$$\lambda' - \lambda = \frac{h}{m_e c}(1 - \cos\theta)$$

Compton Wavelength of Electron

$$\lambda_C = \frac{h}{m_e c} \approx 2.43 \times 10^{-12}\ \text{m}$$

De Broglie Hypothesis (Matter Waves)

De Broglie Wavelength

$$\lambda = \frac{h}{p} = \frac{h}{mv}$$

From Kinetic Energy

$$\lambda = \frac{h}{\sqrt{2m \cdot KE}}$$

For Particle Accelerated Through Potential $V$

$$\lambda = \frac{h}{\sqrt{2meV}}$$

Wave Functions & Probability

Probability Density (Born Interpretation)

$$P(x,t),dx = |\Psi(x,t)|^2,dx = \Psi^*(x,t)\Psi(x,t),dx$$

Normalization Condition

$$\int_{-\infty}^{\infty} |\Psi(x,t)|^2,dx = 1$$

Requirements for Valid Wave Functions

Single-valued
Continuous
Finite
Square-integrable

Heisenberg Uncertainty Principle

Position–Momentum

$$\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$$

Energy–Time

$$\Delta E \cdot \Delta t \geq \frac{\hbar}{2}$$

Practical Form

$$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}, \quad \Delta E \cdot \Delta t \geq \frac{h}{4\pi}$$

Minimum Uncertainty Estimates

$$\Delta p \geq \frac{\hbar}{2\Delta x}, \quad \Delta v \geq \frac{\hbar}{2m,\Delta x}$$

Schrödinger Equation

Time-Dependent (TDSE)

$$i\hbar\frac{\partial\Psi(x,t)}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x)\Psi(x,t)$$

Separation of Variables

$$\Psi(x,t) = \psi(x) \cdot e^{-iEt/\hbar}$$

Time-Independent (TISE)

$$-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)$$

Hamiltonian Form

$$\hat{H}\psi(x) = E\psi(x), \quad \hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)$$

Particle in a 1D Infinite Square Well (1D Box)

Potential

$$V(x) = \begin{cases} 0 & 0 < x < L \ \infty & \text{otherwise} \end{cases}$$

Boundary Conditions

$$\psi(0) = 0, \quad \psi(L) = 0$$

Quantized Wave Number

$$k_n = \frac{n\pi}{L}, \quad n = 1, 2, 3, \dots$$

Energy Eigenvalues

$$E_n = \frac{\hbar^2 k_n^2}{2m} = \frac{n^2 h^2}{8mL^2} = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$$

Zero-Point Energy

$$E_1 = \frac{h^2}{8mL^2} > 0$$

Normalized Wave Functions

$$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$$

Probability Density

$$|\psi_n(x)|^2 = \frac{2}{L}\sin^2\left(\frac{n\pi x}{L}\right)$$

Nodes

State $\psi_n$ has $(n-1)$ nodes inside the box.

Energy Level Spacing

$$\Delta E = E_{n+1} - E_n = \frac{(2n+1)h^2}{8mL^2}$$

$$\frac{E_{n+1} - E_n}{E_n} = \frac{2n+1}{n^2} \approx \frac{2}{n} \quad \text{as } n \to \infty$$

Energy Transitions & Photon Emission

Photon Energy from Transition

$$\Delta E = E_{n_i} - E_{n_f} = \frac{h^2}{8mL^2}\left(n_i^2 - n_f^2\right)$$

Wavelength of Emitted Photon

$$\lambda = \frac{hc}{\Delta E}$$

Quick-Reference Summary Tables

Full-Course Formula Index

Topic	Key Formula
Coulomb's Law	$F = \dfrac{kQq}{r^2}$
Electric Field (point charge)	$E = \dfrac{kQ}{r^2}$
Capacitance	$C = \dfrac{Q}{\Delta V} = \dfrac{A\varepsilon_0}{d}$
Ohm's Law	$V = IR$
EMF & Terminal Voltage	$V_{\text{terminal}} = \varepsilon - Ir$
Resistors in Series	$R_{\text{eq}} = \sum R_i$
Resistors in Parallel	$\dfrac{1}{R_{\text{eq}}} = \sum \dfrac{1}{R_i}$
Power (DC)	$P = IV = I^2R = \dfrac{V^2}{R}$
RMS Values	$V_{\text{rms}} = \dfrac{V_{\text{max}}}{\sqrt{2}}$
Reactances	$X_L = 2\pi f L$, $X_C = \dfrac{1}{2\pi f C}$
RLC Impedance	$Z = \sqrt{R^2 + (X_L - X_C)^2}$
Resonance Frequency	$f_0 = \dfrac{1}{2\pi\sqrt{LC}}$
Power Factor	$\text{PF} = \cos\phi = \dfrac{R}{Z}$
Straight Wire $B$-Field	$B = \dfrac{\mu_0 I}{2\pi r}$
Solenoid $B$-Field	$B = \mu_0 n I$
Lorentz Force	$\vec{F} = q\vec{v} \times \vec{B}$
Circular Motion in $B$	$r = \dfrac{mv}{qB}$
Faraday's Law	$\varepsilon = -N\dfrac{d\Phi_B}{dt}$
Motional EMF	$\varepsilon = Blv$
Self-Inductance	$\mathcal{E} = -L\dfrac{dI}{dt}$
Transformer Ratio	$\dfrac{V_s}{V_p} = \dfrac{N_s}{N_p}$
Diode Forward Bias	$V_D = 0.7\ \text{V}$ (Si)
BJT Current Gain	$\beta = \dfrac{I_C}{I_B}$
Op-Amp Inverting	$V_{\text{out}} = -\dfrac{R_f}{R_1}V_{\text{in}}$
Bohr Energy Levels	$E_n = -\dfrac{13.6}{n^2}\ \text{eV}$
Rydberg Formula	$\dfrac{1}{\lambda} = R_H\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)$
Mass Defect	$\Delta m = Zm_p + Nm_n - m_N$
Binding Energy	$E_B = \Delta m \times 931.5\ \text{MeV}$
Radioactive Decay	$N(t) = N_0 e^{-\lambda t}$
Half-Life	$T_{1/2} = \dfrac{0.693}{\lambda}$
Photoelectric Effect	$KE_{\max} = hf - \phi$
De Broglie Wavelength	$\lambda = \dfrac{h}{p}$
Heisenberg Uncertainty	$\Delta x , \Delta p \geq \dfrac{\hbar}{2}$
1D Box Energy	$E_n = \dfrac{n^2 h^2}{8mL^2}$
Stefan-Boltzmann	$\dfrac{P}{A} = \sigma T^4$
Wien's Law	$\lambda_{\max} = \dfrac{b}{T}$

Compiled from all FAD1022 lecture notes, tutorials, and concept sheets. No formula omitted. Good luck with finals!