Materials Science and Engineering/Equations/Physics

Coulomb's Law
$$F = k \frac{Q_1 Q_2}{r^2}$$ $$F = \frac{1}{4 \pi \epsilon_0} \frac{Q_1 Q_2}{r^2}$$

The Electric Field
$$\mathbf E = \frac{ \mathbf F}{q}$$

$$dE = \frac{1}{4 \pi \epsilon_0} \frac{dQ}{r^2}$$

Superposition Principle
$$\mathbf E = \mathbf E_1 + \mathbf E_2 + \cdots$$

Electric Flux
$$\Phi_E = \int \mathbf E \cdot d \mathbf A$$

Gauss's Law
$$\oint \mathbf \cdot d \mathbf A = \frac{Q_{encl}}{\epsilon_0}$$

Electrical Potential
$$V_a = \frac{U_a}{q}$$

Relation between Electric Potential and Electric Field
$$U_b - U_a = \int_a^b \mathbf F \cdot d \mathbf l$$

$$V_{ba} = V_b - V_a = - \int_a^b \mathbf E \cdot d \mathbf l$$

Electrical Potential Due to Point Charges
$$E = \frac{1}{4 \pi \epsilon} \frac{Q}{r^2}$$

$$V = \frac{a}{4 \pi \epsilon_0} \frac{Q}{r}$$

Potential Due to Charge Distributions
$$V = \frac{1}{4 \pi \epsilon_0} \int \frac{dq}{r}$$

E Determined from V
$$C = \frac{Q}{V_{ba}} = \epsilon_0 \frac{A}{d}$$

Electrical Energy Storage
$$W = \int_0^Q V dq = \frac{1}{C} \int_0^Q q dq = \frac{Q^2}{C}$$

$$U = \frac{1}{2} CV^2 = \frac{1}{2} \left ( \frac{\epsilon_0 A}{d} \right ) (E^2 d^2)$$ $$U = \frac{1}{2} \epsilon_0 E^2 A d$$ $$u = \mbox{energy density} = \frac{1}{2} \epsilon_0 E^2$$

Dielectrics
$$C = K C_0\;$$ $$C = K \epsilon_0 \frac{A}{d}$$ $$\epsilon = K \epsilon_0\;$$ $$C = \epsilon \frac{A}{d}$$

$$u = \frac{1}{2} K \epsilon_0 E^2 = \frac{1}{2} \epsilon E^2$$

$$Q = K Q_0\;$$

$$V = \frac{V_0}{K}$$

$$E_0 = \frac{V_0}{d}$$

$$E = E_D = \frac{V}{d} = \frac{V_0}{Kd}$$

$$E_D = \frac{E_0}{K}$$

$$E_D = E_0 - E_{ind}\;$$ $$E_D = \frac{E_0}{K}$$

$$E_{ind} = E_0 \left ( 1 - \frac{1}{K} \right )$$

$$E_0 = \frac{\sigma}{\epsilon_0}$$

$$\sigma = \frac{Q}{A}$$

$$E_{ind} = \frac{\sigma_{ind}}{\epsilon_0}$$

$$E_{ind} = E_0 \left (1 - \frac{1}{K} \right )$$

$$\sigma_{ind} = \sigma \left (1 - \frac{1}{K} \right )$$

$$Q_{ind} = Q \left (1 - \frac{1}{K} \right )$$

Electric Current
$$I = \frac{dQ}{dt}$$

Ohm's Law
$$R = \frac{V}{I}$$

Resistivity
$$R = \rho \frac{l}{A}$$

$$\sigma = \frac{1}{\rho}$$

Electric Power
$$P = \frac{dU}{dt} = \frac{dq}{dt} V$$

$$P = IV\;$$

Current Density and Drift Velocity
$$I = \int \mathbf j \cdot d \mathbf A$$

$$\Delta Q = ( \mbox{no. of charges}, N) \times (\mbox{charge per particle})$$

$$\Delta Q = (nV)(-e)\;$$

$$\Delta Q = - (n A v_d \Delta t)(e)\;$$

$$I = \frac{\Delta Q}{\Delta t} = -neAv_d$$

$$\mathbf j = -ne \mathbf v_d$$

$$\mathbf j = \sum_i n_i q_i \mathbf v_{di}$$ $$I = \sum_i n_i q_i v_{di} A$$

$$\mathbf j = \sigma \mathbf E = \frac{1}{\rho} \mathbf E$$

Force on Electric Current in a Magnetic Field
$$\mathbf F = I \mathbf l \times \mathbf B$$

$$d \mathbf F = I d \mathbf l \times \mathbf B$$

Force on Moving Charge in Magnetic Field
$$\mathbf F = q \mathbf v \times \mathbf B$$

Hall Effect
Electric field due to the separation of charge is the Hall field, $$\mathbf E_H$$

In equilibrium, the force from the electric field is balanced by the magnetic force $$e v_d B$$

$$e E_H = e v_d B\;$$

Magnetic Field Due to Straight Wire
$$B = \frac{ \mu_0 }{2 \pi} \frac{I}{r}$$

Force between Two Parallel Wires
$$B_1 = \frac{\mu_0}{2 \pi} \frac{I_1}{d}$$

Ampere's Law
$$\oint \mathbf B \cdot d \mathbf l = \mu_0 I_{encl}$$

Biot-Savart Law
$$d \mathbf B = \frac{\mu_0 I}{4 \pi} \frac{d \mathbf l \times \mathbf \hat{r}}{r^2}$$

Magnetic Fields in Magnetic Materials
$$B = \mu n I\;$$

Paramagnetism
Relative permeability:

$$K_m = \frac{\mu}{\mu_0}$$

Magnetic susceptibility:

$$\Chi_m = K_m - 1\;$$

Magnetization vector, M:

$$\mathbf M = \frac{\mathbf \mu}{V}$$

Curie's law:

$$M = C \frac{B}{T}$$

Faraday's Law of Induction
$$E = - \frac{d \Phi_B}{dt}$$ $$\oint \mathbf E \cdot d \mathbf l = - \frac{d \Phi_B}{dt}$$

Ampere's Law
$$\oint \mathbf B \cdot d \mathbf l = \mu_0 I_{encl} + \mu_0 \epsilon_0 \frac{d \Phi_E}{dt}$$

Gauss's Law of Magnetism
$$\Phi_B = \mathbf B \cdot d \mathbf A$$ $$\Phi_B = \oint \mathbf B \cdot d \mathbf A$$ $$\oint \mathbf E \cdot d \mathbf A = \frac{Q}{\epsilon_0}$$

Maxwell's Equations
$$\oint \mathbf E \cdot d \mathbf A = \frac{Q}{\epsilon_0}$$ $$\oint \mathbf B \cdot d \mathbf A = 0$$ $$\oint \mathbf E \cdot d \mathbf l = - \frac{d \Phi_B}{dt}$$ $$\oint \mathbf B \cdot d \mathbf I = \mu_0 I + \mu_0 \epsilon_0 \frac{d \Phi_E}{dt}$$

Relation between Wavelength and Frequency
$$\lambda = \frac{c}{f}$$

Relation between Frequency and ω
$$\omega = 2 \pi f\;$$

Poynting Vector
$$\mathbf S = \frac{1}{\mu_0} \left ( \mathbf E \times \mathbf B \right )$$

Index of Refraction
$$n = \frac{c}{\nu}$$

Reflection: Snell's Law
$$n_1 \sin \theta_1 = n_2 \sin \theta_2\;$$

Rayleigh Criterion
$$\theta = \frac{1.22 \lambda}{D}$$

Empirical Formula Proposed by Max Planck
$$I(\lambda, T) = \frac{2 \pi h c^2 \lambda^{-5}}{e^{hc/\lambda k T} - 1}$$

Energy Emitted in Packets or Quanta
$$E = hf\;$$

Wave Nature of Matter
$$\lambda = \frac{h}{p}$$

Heisenberg Uncertainty Principle
$$(\Delta x)(\Delta p_x) \ge \frac{h}{2 \pi}$$

One-Dimensional Time-Independent Schrödinger Equation
$$- \frac{\hbar^2}{2m} \frac{d^2 \Psi (x)}{dx^2} + U(x) \Psi (x) = E \Psi (x)$$

Time-Dependent Schrödinger Equation
$$- \frac{\hbar^2}{2m} \frac{\partial^2 \Psi (x,t)}{\partial x^2} + U(x) \Psi (x,t) = i \hbar \frac{\partial \Psi(x,t)}{\partial t}$$

$$\Psi (x,t) = \psi (x) e^{- i \left ( \frac{E}{\hbar} \right ) t}$$

Solution to Schrödinger Equation - Free Particle
$$\psi = A \sin kx + B \cos kx\;$$

$$k = \sqrt{\frac{2mE}{\hbar^2}}$$

Solution to Schrödinger Equation - Infinitely Deep Square Well
$$\Psi_n = \sqrt{ \frac{2}{L} } \sin \left ( \frac{n \pi}{L} x \right )$$