PHY1008 Engineering Physics Formula Sheet and Comprehensive Study Guide

Module 5: Magnetic Materials

• Basic Magnetic Relations

  • Magnetic Susceptibility ($\chi_m$): Defined as the ratio of magnetisation to field intensity, given by the formula $\chi_m = \frac{M}{H}$. Here, $M$ is the magnetisation ($A/m$) and $H$ is the magnetic field intensity ($A/m$).
  • Relative Permeability ($\mu_r$): Represents the ratio of the permeability of a specific medium to the permeability of free space, calculated as $\mu_r = 1 + \chi_m$.
  • Absolute Permeability ($\mu$): Calculated as the product of the permeability of free space and the relative permeability, $\mu = \mu_0 \times \mu_r$. The value of $\mu_0$ is $4\pi \times 10^{-7}\,H/m$.
  • Magnetisation ($M$): Can be expressed as the product of susceptibility and field intensity: $M = \chi_m \times H$.
  • Magnetic Flux Density ($B$): Measured in Tesla ($T$) or Webers per square meter ($Wb/m^2$). It is given by $\mathbf{B} = \mu_0(H + M) = \mu_0 \mu_r H$.

• Orbital Electron — Atomic Magnetism

  • Frequency of Revolution ($f$): The reciprocal of the time period, $f = \frac{1}{T}$, where $T$ is measured in seconds ($s$).
  • Orbital Velocity ($v$): Calculated based on the orbit circumference and time period as $v = \frac{2\pi r}{T}$, where $r$ is the orbital radius.
  • Orbital Current ($I$): The current generated by the revolving electron is $I = e \times f$, where $e = 1.6 \times 10^{-19}\,C$.
  • Magnetic Moment ($m$): Defined as the product of current and the area of the orbit, $m = I \times A = I \times \pi r^2$, where $A$ is the area of the orbit.
  • Bohr Magneton ($\mu_B$): The fundamental unit of magnetic moment, with a value of $\mu_B = 9.274 \times 10^{-24}\,A \cdot m^2$.
  • Magnetic Flux Density ($B$) at the Centre of a Circular Orbit: Given by $B = \frac{\mu_0 I}{2r}$.

• Magnetic Properties of Circular Loops (N turns)

  • Magnetic Moment ($m$): For a loop with multiple turns, $m = N \times I \times A$. Here, $N$ is the number of turns and $A = \pi r^2$.
  • Magnetic Flux Density ($B$) at the Centre: Calculated as $B = \frac{\mu_0 N I}{2r}$.
  • Magnetic Flux ($\Phi$): Measured in Webers ($Wb$), defined as $\Phi = B \times A$.

• Curie's Law for Paramagnetism

  • Curie’s Law: The susceptibility is inversely proportional to the absolute temperature, $\chi_m = \frac{C}{T}$, where $C$ is the Curie constant and $T$ is temperature in Kelvin ($K$).
  • Temperature Scaling: For different states, the relationship follows $\chi_1 T_1 = \chi_2 T_2$.

• Saturation Magnetisation

  • Number Density of Atoms ($n$): Calculated using the formula $n = \frac{\rho \times N_A}{M_{At}}$, where $\rho$ is density, $N_A = 6.022 \times 10^{23}/mol$ (Avogadro's number), and $M_{At}$ is the atomic weight.
  • Saturation Magnetisation ($M_s$): Given by $M_s = n \times f \times \mu_B$, where $f$ represents the number of Bohr magnetons per atom.

Module 4: Semiconductors

• Intrinsic Carrier Concentration

  • Intrinsic Carrier Density ($n_i$): The formula is $n_i = 4.83 \times 10^{21} \times T^{3/2} \times \exp(\frac{-E_g}{2kT})$. This specific version is valid when the effective masses are equal to the rest mass ($m_e^* = m_h^* = m_0$).
  • Mass Action Law: In thermal equilibrium, the product of electron and hole concentrations is constant, $n \times p = n_i^2$.
  • General $n_i$ from Density of States: $n_i^2 = N_C \times N_V \times \exp(\frac{-E_g}{kT})$, where $N_C$ and $N_V$ are the effective densities of states in the conduction and valence bands respectively.

• Conductivity and Resistivity

  • Intrinsic Conductivity ((\sigma)): $\sigma = n_i \times e \times (\mu_e + \mu_h)$, where $e = 1.6 \times 10^{-19}\,C$.
  • Extrinsic Conductivity ($n$-type): $\sigma \approx N_d \times e \times \mu_e$, where $N_d$ is the donor density and $N_d \gg n_i$.
  • Extrinsic Conductivity ($p$-type): $\sigma \approx N_a \times e \times \mu_h$, where $N_a$ is the acceptor density and $N_a \gg n_i$.
  • Resistivity ((\rho)): Calculated as $\rho = \frac{1}{\sigma}$ and measured in $\Omega \cdot m$.
  • Resistance of a Rod ($R$): Calculated as $R = \frac{\rho \times L}{A}$, where $L$ is the length and $A$ is the cross-sectional area.
  • Mobility ((\mu)): Defined as $\mu = \frac{v_d}{E}$, where $v_d$ is the drift velocity and $E$ is the electric field in $V/m$.

• Temperature Dependence and Fermi Level

  • Two-Temperature Relation: $\ln(\frac{\sigma_2}{\sigma_1}) = (\frac{E_g}{2k}) \times (\frac{1}{T_1} - \frac{1}{T_2})$. This allows the calculation of the band gap ($E_g$) from two resistivity or conductivity measurements.
  • Fermi Level Position ($E_F$) for Equal Effective Masses: $E_F = E_v + \frac{E_g}{2}$. The Fermi level sits exactly at the mid-gap.
  • Fermi Level Position ($E_F$) for Unequal Effective Masses: $E_F = E_v + \frac{E_g}{2} + (\frac{3kT}{4}) \times \ln(\frac{m_h^*}{m_e^*})$.
  • Electron Concentration from $E_F$: $n_0 = N_C \times \exp(\frac{-(E_C - E_F)}{kT})$.
  • Hole Concentration from $E_F$: $p_0 = N_V \times \exp(\frac{-(E_F - E_V)}{kT})$.

• Hall Effect

  • Hall Coefficient for $n$-type ($R_H$): $R_H = \frac{-1}{n \times e}$; the negative sign indicates electrons are the charge carriers.
  • Hall Coefficient for $p$-type ($R_H$): $R_H = \frac{+1}{p \times e}$.
  • Hall Voltage ($V_H$): $V_H = \frac{R_H \times I \times B}{t}$, where $t$ is the sample thickness in the direction of the magnetic field $B$.
  • Hall Electric Field ($E_H$): $E_H = R_H \times J \times B$, where $J$ is the current density ($A/m^2$).
  • Carrier Density: $n = \frac{1}{|R_H| \times e}$.
  • Mobility from Hall Measurements: $\mu = |R_H| \times \sigma = \frac{|R_H|}{\rho}$.
  • Drift Velocity ($v_d$): $v_d = \frac{J}{n \times e}$.

• PN Junction and Optoelectronics

  • Built-in Potential ($V_0$): $V_0 = (\frac{kT}{e}) \times \ln(\frac{N_a \times N_d}{n_i^2})$. The term $\frac{kT}{e} = 0.02585\,V$ at $300\,K$.
  • LED Emission Wavelength ((\lambda)): $\lambda = \frac{hc}{E_g}$. The product $hc$ is taken as $1240\,eV \cdot nm$.
  • Solar Cell Fill Factor ($FF$): $FF = \frac{P_{max}}{I_{sc} \times V_{oc}}$, where $P_{max}$ is the maximum power obtained from the I–V curve.
  • Solar Cell Efficiency ((\eta)): $\eta = \frac{P_{max}}{P_{in}} = \frac{P_{max}}{G \times A}$, where $G$ is the irradiance ($W/m^2$).
  • Thermistor Resistance ($R$): $R = R_0 \times \exp(\frac{E_g}{2kT})$, where $R_0$ is a pre-exponential constant.

Module 3: Quantum Mechanics

• de Broglie Wave-Particle Duality

  • de Broglie Wavelength ((\lambda)): $\lambda = \frac{h}{p} = \frac{h}{m \times v}$, where $h = 6.626 \times 10^{-34}\,J \cdot s$.
  • Wavelength for Accelerated Particle (charge $q$): $\lambda = \frac{h}{\sqrt{2mqV}}$, where $V$ is the accelerating potential in volts.
  • Quick Formula for Electrons: $\lambda = \frac{12.27}{\sqrt{V}}\,\text{…}$, with $V$ in volts.
  • Wavelength from Kinetic Energy ($E_k$): $\lambda = \frac{h}{\sqrt{2mE_k}}$, where $E_k$ is in Joules.

• Phase and Group Velocity

  • Phase Velocity ($v_p$): The speed of a single frequency wave component, $v_p = \frac{\omega}{k} = \lambda \times f$, where $\omega$ is angular frequency and $k$ is wave number.
  • Group Velocity ($v_g$): The speed of the wave packet envelope, $v_g = \frac{d\omega}{dk}$.
  • General Relation: $v_g = v_p - \lambda \times (\frac{dv_p}{d\lambda})$.
  • Water Gravity Waves: Special case where $v_p = \sqrt{g\lambda}$, then $v_g = \frac{v_p}{2}$.
  • Two Superposed Waves: $v_g = \frac{\Delta \omega}{\Delta k}$ and packet phase velocity $v_p(packet) = \frac{\omega_1 + \omega_2}{k_1 + k_2}$.

• Heisenberg Uncertainty Principle

  • Position-Momentum: $\Delta x \times \Delta p \ge (\frac{h}{2\pi})$, where $\frac{h}{2\pi} = 1.055 \times 10^{-34}\,J \cdot s$.
  • Energy-Time: $\Delta E \times \Delta t \ge (\frac{h}{2\pi})$.
  • Frequency Uncertainty: $\Delta \nu = \frac{\Delta E}{h}$.
  • Velocity Uncertainty: $\Delta v = \frac{\Delta p}{m}$.
  • Minimum Kinetic Energy from Confinement ($KE_{min}$): Calculated as $KE_{min} = \frac{(\Delta p)^2}{2m}$, where $\Delta p = \frac{(\frac{h}{2\pi})}{\Delta x}$.

• Particle in a 1D Infinite Potential Box

  • Allowed Energies ($E_n$): $E_n = \frac{n^2 \pi^2 (\frac{h}{2\pi})^2}{2mL^2} = n^2 E_1$, for $n = 1, 2, 3, \dots$ and box width $L$.
  • Ground State Energy ($E_1$): $E_1 = \frac{\pi^2 (\frac{h}{2\pi})^2}{2mL^2}$.
  • Wave Function ((\psi_n(x))): $\psi_n(x) = \sqrt{\frac{2}{L}} \times \sin(\frac{n\pi x}{L})$ for $0 \le x \le L$.
  • de Broglie Wavelength in Box ((\lambda_n)): $\lambda_n = \frac{2L}{n}$, fulfilling the standing-wave condition.
  • Probability Density: $|\psi_n|^2 = (\frac{2}{L}) \times \sin^2(\frac{n \pi x}{L})$.
  • Probability in Interval [a, b]: $P = \int_a^b |\psi|^2 dx$.
  • Normalisation Condition: $\int_{-\infty}^{+\infty} |\psi|^2 dx = 1$. This condition is used to determine the normalisation constant $A$.

Module 2: Diffraction

• Single-Slit Fraunhofer Diffraction

  • Minima Condition: $a \times \sin(\theta_m) = m \times \lambda$, where $a$ is slit width and $m = \pm 1, \pm 2, \dots$
  • Secondary Maxima: $a \times \sin(\theta) = (2m + 1) \times \frac{\lambda}{2}$.
  • Half Angular Width of Central Maximum: $\sin(\theta) = \frac{\lambda}{a}$.
  • Width of Central Maximum on Screen ($W$): $W = \frac{2\lambda D}{a}$, where $D$ is the slit-to-screen distance.
  • Position of m-th Minimum ($y_m$): $y_m = \frac{m\lambda D}{a}$.

• Double-Slit / Young's Experiment

  • Interference Maxima: $p \times \sin(\theta) = m \times \lambda$, where $p$ is the centre-to-centre slit separation and $m = 0, \pm 1, \pm 2, \dots$
  • Fringe Width ((\beta)): $\beta = \frac{\lambda D}{p}$.
  • Position of m-th Bright Fringe ($y_m$): $y_m = \frac{m \lambda D}{p}$.
  • Missing Orders Condition: $n = \frac{p}{a}$. When this ratio is an integer, specific orders of diffraction are missing.

• Diffraction Grating

  • Grating Equation: $d \times \sin(\theta) = m \times \lambda$, where $d$ is the grating spacing ($d = 1/N_{lines}$ per unit length).
  • Dispersive Power: $\frac{d\theta}{d\lambda} = \frac{m}{d \times \cos(\theta)}$ in units of $rad/m$. To convert to $deg/\text{…}$, multiply by $(\frac{180}{\pi}) \times 10^{-10}$.
  • Resolving Power ($RP$): $RP = \frac{\lambda}{\Delta \lambda} = N \times m$, where $N$ is the total number of lines and $m$ is the order.
  • Smallest Resolvable Change in Wavelength ((\Delta \lambda)): $\Delta \lambda = \frac{\lambda}{RP}$.
  • Highest Visible Order ($m_{max}$): $m_{max} = \lfloor \frac{d}{\lambda} \rfloor$, determined at $\theta = 90^\circ$ where $\sin(\theta) = 1$.
  • Overlapping Orders: Condition for spectral overlap is $m_1 \times \lambda_1 = m_2 \times \lambda_2$.

• Resolving Power of Optical Instruments

  • Rayleigh Criterion: Minimum angular resolution is $\theta_{min} = \frac{1.22\lambda}{D}$, where $D$ is the objective lens diameter.
  • Resolving Power ($RP$): $RP = \frac{D}{1.22\lambda}$.
  • Minimum Resolvable Distance ($d_{min}$): $d_{min} = \theta_{min} \times L$, where $L$ is the distance to the object.

Module 1: Interference

• Thin Film Interference

  • Optical Path Difference ((\Delta)): $\Delta = 2\mu t \times \cos(\theta_r)$, where $\mu$ is refractive index, $t$ is thickness, and $\theta_r$ is the refraction angle.
  • Phase Change on Reflection: A phase change of $+\pi$ (half-wave loss) occurs when reflecting off a denser medium; no phase change occurs when reflecting off a rarer medium.
  • Conditions with One Phase Change (e.g., soap film in air):
    • Bright Fringe: $2\mu t \times \cos(\theta_r) = (m - \frac{1}{2})\lambda$, where $m = 1, 2, \dots$
    • Dark Fringe: $2\mu t \times \cos(\theta_r) = m\lambda$, where $m = 0, 1, 2, \dots$
  • Conditions with No Net Phase Change:
    • Bright Fringe: $2\mu t \times \cos(\theta_r) = m\lambda$.
    • Dark Fringe: $2\mu t \times \cos(\theta_r) = (m - \frac{1}{2})\lambda$.
  • Anti-reflection Coating (Minimum Thickness): $t_{min} = \frac{\lambda}{4\mu_f}$ (assuming one phase change to minimise reflection).

• Newton's Rings

  • n-th Dark Ring Diameter (Air film, reflected light): $D_n^2 = 4nR\lambda$, where $R$ is the radius of curvature of the lens.
  • n-th Bright Ring Diameter: $D_n^2 = (4n + 2)R\lambda$.
  • With Liquid (Index $\mu$): $D_n^2 = \frac{4nR\lambda}{\mu}$.
  • Wavelength ((\lambda)) from Diameters: $\lambda = \frac{D_m^2 - D_n^2}{4(m - n)R}$.
  • Refractive Index of Liquid ((\mu)): $\mu = \frac{(D_m^2 - D_n^2)_{air}}{(D_m^2 - D_n^2)_{liq}}$.
  • Radius of Curvature ($R$): $R = \frac{D_n^2}{4n\lambda}$, using the dark ring diameter.
  • Thickness at n-th Dark Ring ($t$): $t = \frac{n\lambda}{2}$ (derived from $2t = n\lambda$ for conditions with one phase change).

• Young's Double-Slit Experiment (Detailed)

  • Path Difference ((\Delta)): $\Delta = \frac{d \times y}{D} = d \times \sin(\theta)$, where $d$ is slit separation, $y$ is distance from the centre, and $D$ is screen distance.
  • Phase Difference ((\delta)): $\delta = (\frac{2\pi}{\lambda}) \times \Delta$.
  • Fringe Width ((\beta)): $\beta = \frac{\lambda D}{d}$.
  • m-th Bright Fringe Position ($y_m$): $y_m = \frac{m \lambda D}{d}$, for $m = 0, \pm 1, \pm 2, \dots$
  • m-th Dark Fringe Position ($y_m$): $y_m = \frac{(2m - 1) \lambda D}{2d}$.
  • Coincidence of Two Wavelengths: Condition $n_1 \beta_1 = n_2 \beta_2$. Solve for the smallest integers using the LCM of $\beta_1$ and $\beta_2$.

• Intensity in Interference

  • Resultant Intensity ($I$): $I = I_1 + I_2 + 2\sqrt{I_1 I_2} \times \cos(\delta)$.
  • Maximum Intensity ($I_{max}$): $I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2$, occurring at $\delta = 0, 2\pi, \dots$
  • Minimum Intensity ($I_{min}$): $I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2$, occurring at $\delta = \pi, 3\pi, \dots$
  • Source Ratio from $I_{max}/I_{min}$: $\sqrt{\frac{I_1}{I_2}} = \frac{\sqrt{r} + 1}{\sqrt{r} - 1}$, where $r = \frac{I_{max}}{I_{min}}$.
  • Amplitude Ratio ($A_{max}/A_{min}$): $\frac{A_{max}}{A_{min}} = \frac{A_1 + A_2}{|A_1 - A_2|}$, noting that amplitude $A \propto \sqrt{I}$.

• Michelson Interferometer

  • Wavelength ((\lambda)) from Fringe Count: $\lambda = \frac{2 \times \Delta d}{N}$, where $\Delta d$ is the mirror displacement and $N$ is the number of fringes counted.

Physical Constants

• Fundamental Constants

  • Planck's constant ($h$): $6.626 \times 10^{-34}\,J \cdot s$
  • Reduced Planck's constant ($\frac{h}{2\pi}$): $1.055 \times 10^{-34}\,J \cdot s$
  • Speed of light ($c$): $3 \times 10^8\,m/s$
  • Electron charge ($e$): $1.6 \times 10^{-19}\,C$
  • Electron rest mass ($m_e$): $9.11 \times 10^{-31}\,kg$
  • Proton mass ($m_p$): $1.67 \times 10^{-27}\,kg$
  • Neutron mass ($m_n$): $1.674 \times 10^{-27}\,kg$
  • Permeability of free space ($\mu_0$): $4\pi \times 10^{-7}\,H/m$
  • Boltzmann constant ($k_B$): $8.617 \times 10^{-5}\,eV/K = 1.381 \times 10^{-23}\,J/K$
  • Avogadro number ($N_A$): $6.022 \times 10^{23}/mol$
  • Bohr magneton ($\mu_B$): $9.274 \times 10^{-24}\,A \cdot m^2$

• Derived values and Conversions

  • Thermal energy $kT$ at $300\,K$: $0.02585\,eV \approx 0.026\,eV$
  • $hc$ product: $1240\,eV \cdot nm = 1.988 \times 10^{-25}\,J \cdot m$
  • Energy conversion: $1\,eV = 1.6 \times 10^{-19}\,J$
  • Length conversion (Angstrom): $1\,\text{…} = 10^{-10}\,m$
  • Length conversion (Nanometer): $1\,nm = 10^{-9}\,m$

• Note: All formulas used in this sheet assume SI units unless specified otherwise. $\frac{h}{2\pi}$ is the reduced Planck constant (h-bar).