Formula Sheet — Tutorial 13: Electronics + Atomic Physics

1. BJT Transistor — Voltage Divider Bias

Approximate Analysis Condition

$$\beta R_E \geq 10 R_B$$

Variable Meaning Units
$\beta$ DC current gain unitless
$R_E$ Emitter resistor $\Omega$
$R_B$ Base resistor (or equivalent) $\Omega$

Base Voltage (Voltage Divider)

$$V_B = \frac{R_2}{R_1 + R_2} V_{CC}$$

Variable Meaning Units
$V_B$ Base voltage V
$R_1, R_2$ Voltage divider resistors $\Omega$
$V_{CC}$ Collector supply voltage V

Emitter Voltage

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

Variable Meaning Units
$V_E$ Emitter voltage V
$V_{BE}$ Base-emitter voltage ($\approx 0.7 \text{ V}$ for Si) V

Emitter Current

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

Variable Meaning Units
$I_E$ Emitter current A
$I_C$ Collector current A

Collector-Emitter Voltage

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

Variable Meaning Units
$V_{CE}$ Collector-emitter voltage V
$R_C$ Collector resistor $\Omega$

DC Current Gain

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

2. Operational Amplifiers (Op-Amps)

Inverting Amplifier

$$A_v = -\frac{R_f}{R_1}$$

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

Variable Meaning Units
$A_v$ Voltage gain unitless
$R_f$ Feedback resistor $\Omega$
$R_1$ Input resistor $\Omega$
$V_{out}$ Output voltage V
$V_{in}$ Input voltage V

Current Through Input Resistor (Inverting)

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

Variable Meaning Units
$I$ Current through resistor A

Non-Inverting Amplifier

$$A_v = 1 + \frac{R_f}{R_1}$$

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

Variable Meaning Units
$A_v$ Voltage gain unitless
$R_f$ Feedback resistor $\Omega$
$R_1$ Input resistor $\Omega$

3. Bohr Model of the Hydrogen Atom

Coulomb Force (Centripetal Force)

$$\frac{ke^2}{r_n^2} = \frac{mv^2}{r_n}$$

Variable Meaning Units
$k$ Coulomb constant $8.99 \times 10^9 \text{ N m}^2/\text{C}^2$
$e$ Elementary charge $1.602 \times 10^{-19} \text{ C}$
$r_n$ Orbital radius of $n$-th level m
$m$ Electron mass $9.11 \times 10^{-31} \text{ kg}$
$v$ Electron orbital speed m/s

Quantized Angular Momentum (Bohr's Postulate)

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

Variable Meaning Units
$L$ Angular momentum kg m$^2$/s
$n$ Principal quantum number unitless ($n = 1, 2, 3, ...$)
$\hbar$ Reduced Planck constant $1.055 \times 10^{-34} \text{ J s}$
$h$ Planck constant $6.626 \times 10^{-34} \text{ J s}$

Orbital Radius of $n$-th Level

$$r_n = \frac{n^2 \hbar^2}{mke^2} = n^2 a_0$$

where $a_0 = 0.529 \times 10^{-10} \text{ m} = 0.529 \text{ \AA}$ (Bohr radius)

Variable Meaning Units
$r_n$ Orbital radius m
$a_0$ Bohr radius m

Total Energy of $n$-th Level

$$E_n = -\frac{mk^2 e^4}{2\hbar^2} \left(\frac{1}{n^2}\right)$$

$$E_n = -13.6 \text{ eV} \left(\frac{1}{n^2}\right)$$

Variable Meaning Units
$E_n$ Total energy of $n$-th level eV (or J)
$n$ Principal quantum number unitless

Kinetic Energy

$$KE = \frac{1}{2}mv^2 = \frac{ke^2}{2r_n} = -E_n$$

Potential Energy

$$PE = -\frac{ke^2}{r_n} = 2E_n$$

Energy of Photon Emitted/Absorbed

$$\Delta E = E_i - E_f = hf = \frac{hc}{\lambda}$$

Variable Meaning Units
$\Delta E$ Energy difference J or eV
$E_i$ Initial energy level J or eV
$E_f$ Final energy level J or eV
$f$ Photon frequency Hz
$\lambda$ Photon wavelength m

Rydberg Formula (Wavelength of Spectral Lines)

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

Variable Meaning Units
$R_H$ Rydberg constant $1.097 \times 10^7 \text{ m}^{-1}$
$n_i$ Initial quantum number unitless
$n_f$ Final quantum number unitless

Spectral Series

Series Transition $n_f$
Lyman UV 1
Balmer Visible 2
Paschen Infrared 3
Brackett Infrared 4
Pfund Infrared 5

4. Summary Table of Key Constants

Constant Symbol Value
Planck constant $h$ $6.626 \times 10^{-34} \text{ J s}$
Reduced Planck constant $\hbar$ $1.055 \times 10^{-34} \text{ J s}$
Coulomb constant $k$ $8.99 \times 10^9 \text{ N m}^2/\text{C}^2$
Elementary charge $e$ $1.602 \times 10^{-19} \text{ C}$
Electron mass $m_e$ $9.11 \times 10^{-31} \text{ kg}$
Bohr radius $a_0$ $0.529 \times 10^{-10} \text{ m}$
Rydberg constant $R_H$ $1.097 \times 10^7 \text{ m}^{-1}$
Speed of light $c$ $3.0 \times 10^8 \text{ m/s}$

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