Formula sheet

Engineering Physics Formula Sheet

Engineering physics formulas for radiation dose, plasma, piezoelectric charge, Seebeck voltage, photodiodes, heat flux, Knudsen number, resonance, and uncertainty.

Branch: Engineering Physics
Content: Formula sheet
Updated: May 29, 2026
Revision: v1.0.11 · reviewed

This formula sheet collects common first-pass relationships used when turning physical effects into engineered sensors, actuators, optical devices, heat-transfer measurements, rarefied-flow models, and uncertainty estimates. Use it for screening and review; detailed design still requires material data, calibration, boundary-condition checks, and validation in the intended operating regime.

State the physical regime before using a formula: temperature range, pressure, wavelength, frequency, geometry, material direction, bias condition, mounting, and whether the model assumes linearity or continuum behaviour.

Radiation dose screening

Absorbed dose:

\displaystyle D=\frac{E}{m}

where $E$ is deposited energy and $m$ is irradiated mass.

Dose rate:

\displaystyle \dot{D}=\frac{D}{t}

Equivalent dose:

H=w_R D

where $w_R$ is radiation weighting factor.

Radioactive decay:

N(t)=N_0e^{-\lambda t}

Half-life:

\displaystyle t_{1/2}=\frac{\ln 2}{\lambda}

Dose calculations must state geometry, shielding, exposure time, particle or photon energy, detector calibration, and whether absorbed or equivalent dose is being reported.

Plasma screening

Electron plasma frequency:

\displaystyle \omega_{pe}=\sqrt{\frac{n_e e^2}{\epsilon_0 m_e}}

Debye length:

\displaystyle \lambda_D=\sqrt{\frac{\epsilon_0 k_BT_e}{n_e e^2}}

Charged-particle gyrofrequency:

\displaystyle \omega_c=\frac{|q|B}{m}

Plasma screening should state species, density, temperature, magnetic field, collision regime, sheath effects, and whether the plasma can be treated as quasi-neutral.

Piezoelectric charge and voltage

Simplified direct piezoelectric charge relation:

Q=dF

where $Q$ is generated charge, $d$ is a piezoelectric coefficient, and $F$ is applied force along the relevant material direction.

Open-circuit voltage estimate:

\displaystyle V=\frac{Q}{C}

where $C$ is element capacitance.

Capacitive impedance:

\displaystyle Z_C=\frac{1}{j\omega C}

Piezoelectric response depends on direction, preload, frequency, temperature, leakage, cable capacitance, and amplifier input impedance.

Piezoelectric resonance

Undamped natural frequency:

\displaystyle \omega_n=\sqrt{\frac{k}{m}}

Frequency in hertz:

\displaystyle f_n=\frac{\omega_n}{2\pi}

Quality factor:

\displaystyle Q=\frac{f_0}{\Delta f}

where $f_0$ is resonant frequency and $\Delta f$ is bandwidth between half-power points.

High Q gives strong selectivity but narrow bandwidth and longer settling time.

Seebeck voltage

Local approximation:

V\approx S\Delta T

where $S$ is Seebeck coefficient and $\Delta T$ is temperature difference.

For a temperature-dependent coefficient:

V=\int_{T_{ref}}^{T_{hot}} S(T)\,dT

Thermocouple measurement requires reference-junction compensation. Unintended metal junctions in temperature gradients can create additional Seebeck voltages.

Heat flux

Heat flux:

\displaystyle q''=\frac{\dot{Q}}{A}

One-dimensional conduction:

\displaystyle q''=-k\frac{dT}{dx}

Finite-difference conduction estimate:

\displaystyle q''\approx k\frac{T_1-T_2}{L}

Convective heat flux:

q''=h(T_s-T_\infty)

Radiative heat flux between an ideal surface and surroundings:

q''=\epsilon\sigma(T_s^4-T_{sur}^4)

Use absolute temperature in radiation calculations.

Photodiode responsivity

Photodiode current:

I_p=R_\lambda P_{opt}

where $R_\lambda$ is responsivity and $P_{opt}$ is optical power.

Responsivity from quantum efficiency:

\displaystyle R_\lambda=\eta\frac{q\lambda}{hc}

where $\eta$ is quantum efficiency, $q$ is elementary charge, $\lambda$ is wavelength, $h$ is Planck constant, and $c$ is speed of light.

Optical systems must also check dark current, shot noise, amplifier noise, saturation, wavelength range, temperature, and optical alignment.

Quantum efficiency

External quantum efficiency:

\displaystyle \eta=\frac{\text{collected carriers per second}}{\text{incident photons per second}}

Photon energy:

\displaystyle E_{ph}=\frac{hc}{\lambda}

Photon flux from optical power:

\displaystyle \Phi=\frac{P_{opt}}{E_{ph}}=\frac{P_{opt}\lambda}{hc}

Quantum efficiency can depend strongly on wavelength, absorption depth, surface reflection, recombination, bias, temperature, and device structure.

Laser diode screening

Optical power above threshold can be approximated as:

P_{opt}\approx \eta_s(I-I_{th})

where $\eta_s$ is slope efficiency, $I$ is drive current, and $I_{th}$ is threshold current.

Electrical input power:

P_{elec}=VI

Wall-plug efficiency:

\displaystyle \eta_{wp}=\frac{P_{opt}}{P_{elec}}

Laser diodes require current control, thermal control, electrostatic protection, optical feedback management, and safety review.

Knudsen number

Knudsen number:

\displaystyle Kn=\frac{\lambda}{L}

where $\lambda$ is molecular mean free path and $L$ is characteristic length.

Mean free path for an ideal gas can be estimated as:

\displaystyle \lambda=\frac{k_BT}{\sqrt{2}\pi d^2p}

where $k_B$ is Boltzmann constant, $T$ is absolute temperature, $d$ is molecular diameter, and $p$ is pressure.

Common regime screening:

Kn<0.01 \quad \text{continuum}

0.01<Kn<0.1 \quad \text{slip flow}

0.1<Kn<10 \quad \text{transitional}

Kn>10 \quad \text{free molecular}

The characteristic length must match the local physics being modelled.

Reynolds, Mach, and Nusselt numbers

Reynolds number:

\displaystyle Re=\frac{\rho vL}{\mu}

Mach number:

\displaystyle M=\frac{v}{a}

Nusselt number:

\displaystyle Nu=\frac{hL}{k}

These dimensionless numbers help decide whether flow is viscous, compressible, convective, or heat-transfer dominated in the intended regime.

Electric and magnetic field checks

Uniform electric field estimate:

\displaystyle E=\frac{V}{d}

Magnetic flux:

\Phi_B=BA\cos\theta

Faraday induction:

\displaystyle V=-N\frac{d\Phi_B}{dt}

These ideal relations require geometry and field assumptions. Fringing, saturation, hysteresis, shielding, material permeability, and frequency response may dominate real devices.

Thermal response

First-order thermal response:

T(t)=T_\infty+(T_0-T_\infty)e^{-t/\tau}

Thermal time constant approximation:

\tau=R_{th}C_{th}

where $R_{th}$ is thermal resistance and $C_{th}$ is thermal capacitance.

Thermal lag matters in sensors, packages, batteries, laser diodes, thermocouples, and heat-flux measurements.

Signal and noise

Power signal-to-noise ratio:

\displaystyle SNR=\frac{P_{signal}}{P_{noise}}

Decibel form:

SNR_{dB}=10\log_{10}(SNR)

For equal-impedance amplitude ratios:

\displaystyle SNR_{dB}=20\log_{10}\left(\frac{A_{signal}}{A_{noise}}\right)

Measurement bandwidth must be stated. Noise integrated over a wider bandwidth is usually larger.

Uncertainty

Mean value:

\displaystyle \bar{x}=\frac{1}{N}\sum_{i=1}^{N}x_i

Sample standard deviation:

\displaystyle s=\sqrt{\frac{1}{N-1}\sum_{i=1}^{N}(x_i-\bar{x})^2}

Combined standard uncertainty for independent contributions:

u_c=\sqrt{\sum_i u_i^2}

Expanded uncertainty:

U=ku_c

Systematic errors, calibration bias, environmental coupling, and model-form error should not be hidden inside random uncertainty without justification.

Monte Carlo propagation

For a model:

y=f(x_1,x_2,\ldots,x_n)

Monte Carlo uncertainty propagation samples uncertain inputs from their assigned distributions and evaluates the output distribution:

y_j=f(x_{1,j},x_{2,j},\ldots,x_{n,j})

It is useful when the model is nonlinear, input distributions are non-Gaussian, or uncertainty interactions are hard to linearize.

Practical checklist

Use these formulas with a short engineering-physics checklist:

Define the physical quantity, device boundary, and operating regime.
Check signal magnitude, noise, bandwidth, temperature, and material limits.
Verify model validity with dimensionless numbers or scaling checks.
Include package, mounting, thermal path, optical path, and interface electronics.
Build an uncertainty budget and calibration plan.
Validate under intended temperature, pressure, frequency, field, wavelength, and mechanical conditions.

The equations are useful only inside their assumptions. Engineering physics work should always state the regime and the calibration condition behind each result.

REF

Disciplines