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Introductory

General Physics II — Equation Sheet

Complete equation reference for second-semester introductory physics — electric charge, fields, potential, capacitors, circuits, magnetism, induction, Maxwell's equations, and optics.

16 sections · 92 equations

Electric Charge & Coulomb's Law

Coulomb's Law (magnitude)
F=k∣q1∣∣q2∣r2=14πε0∣q1∣∣q2∣r2F = k\frac{|q_1||q_2|}{r^2} = \frac{1}{4\pi\varepsilon_0}\frac{|q_1||q_2|}{r^2}
Electrostatic force between two point charges q₁ and q₂ separated by distance r. Like charges repel; unlike charges attract.
Coulomb's Constant
k=14πε0≈8.99×109 N⋅m2/C2,ε0≈8.85×10−12 C2/(N⋅m2)k = \frac{1}{4\pi\varepsilon_0} \approx 8.99\times10^9\text{ N·m}^2\text{/C}^2, \qquad \varepsilon_0 \approx 8.85\times10^{-12}\text{ C}^2\text{/(N·m}^2\text{)}
ε₀ is the permittivity of free space. The factor 4πε₀ naturally emerges from Gauss's law in SI units.
Elementary Charge
e=1.602×10−19 Ce = 1.602\times10^{-19}\text{ C}
Magnitude of the charge of one proton (+e) or one electron (−e). All observable free charges are integer multiples of e.
Coulomb's Law (vector)
F12=kq1q2r2r^12\mathbf{F}_{12} = k\frac{q_1 q_2}{r^2}\hat{r}_{12}
Force on charge 2 due to charge 1. r̂₁₂ points from 1 to 2. Sign of q₁q₂ determines attraction (negative) or repulsion (positive).
Superposition of Forces
Fnet=∑iFi=kq∑iqiri2r^i\mathbf{F}_{\rm net} = \sum_i \mathbf{F}_i = kq\sum_i \frac{q_i}{r_i^2}\hat{r}_i
The net force on a charge q from multiple source charges is the vector sum of the individual Coulomb forces. Charges act independently.

Electric Field

Electric Field Definition
E=Fq0⇒F=qE\mathbf{E} = \frac{\mathbf{F}}{q_0} \quad \Rightarrow \quad \mathbf{F} = q\mathbf{E}
E is defined as the force per unit positive test charge q₀. A field maps the force a positive charge would feel at every point in space.
Point Charge Field
E=kqr2r^=q4πε0r2r^\mathbf{E} = k\frac{q}{r^2}\hat{r} = \frac{q}{4\pi\varepsilon_0 r^2}\hat{r}
Electric field at distance r from a point charge q. Points radially outward for q > 0, inward for q < 0.
Electric Dipole Moment
p=qd\mathbf{p} = q\mathbf{d}
Dipole moment vector for two charges ±q separated by displacement d (pointing from − to +). SI unit: C·m.
Field from Infinite Line Charge
E=λ2πε0rE = \frac{\lambda}{2\pi\varepsilon_0 r}
Field at perpendicular distance r from an infinitely long wire with linear charge density λ (C/m). Directed radially outward from the wire.
Field from Infinite Plane Sheet
E=σ2ε0E = \frac{\sigma}{2\varepsilon_0}
Uniform field on each side of an infinite sheet with surface charge density σ (C/m²). Field is perpendicular to the sheet and independent of distance.
Torque on a Dipole in a Field
τ=p×E,U=−p⋅E\boldsymbol{\tau} = \mathbf{p}\times\mathbf{E}, \qquad U = -\mathbf{p}\cdot\mathbf{E}
A dipole in a uniform field experiences a torque tending to align p with E. The potential energy is minimum when p is parallel to E.

Gauss's Law

Electric Flux
ΦE=∫SE⋅dA=EAcos⁡θ(uniform E)\Phi_E = \int_S \mathbf{E}\cdot d\mathbf{A} = EA\cos\theta \quad (\text{uniform }\mathbf{E})
Flux counts the number of field lines passing through a surface. The angle θ is between E and the outward surface normal dA.
Gauss's Law
∮SE⋅dA=Qencε0\oint_S \mathbf{E}\cdot d\mathbf{A} = \frac{Q_{\rm enc}}{\varepsilon_0}
The total electric flux through any closed surface equals the enclosed charge divided by ε₀. Most powerful for problems with spherical, cylindrical, or planar symmetry.
Field Outside a Spherical Charge
E=Q4πε0r2=kQr2(r>R)E = \frac{Q}{4\pi\varepsilon_0 r^2} = k\frac{Q}{r^2} \quad (r > R)
A spherical shell or solid sphere with total charge Q behaves like a point charge at its center for all exterior points.
Field Inside a Conductor
Einside=0,Esurface=σε0E_{\rm inside} = 0, \qquad E_{\rm surface} = \frac{\sigma}{\varepsilon_0}
In electrostatic equilibrium, the interior of a conductor has zero field. Any excess charge resides on the surface; the surface field is σ/ε₀ normal to the surface.
Field Inside a Uniform Sphere
E=Qr4πε0R3=kQR3r(r<R)E = \frac{Qr}{4\pi\varepsilon_0 R^3} = k\frac{Q}{R^3}r \quad (r < R)
Field inside a uniformly charged sphere of radius R grows linearly with r. Only the charge enclosed within radius r contributes.

Electric Potential & Potential Energy

Electric Potential (point charge)
V=kqr=q4πε0rV = k\frac{q}{r} = \frac{q}{4\pi\varepsilon_0 r}
Scalar quantity; V = 0 at r → ∞. Superposition applies: V_total = Σᵢ kqᵢ/rᵢ. Unlike force, add as scalars, not vectors.
Work and Potential Difference
WA→B=q(VA−VB)=−ΔUEW_{A\to B} = q(V_A - V_B) = -\Delta U_E
Work done by the electric force moving charge q from A to B. The potential difference V_A − V_B = W/q is independent of path (conservative field).
Electric Potential Energy (two charges)
U=kq1q2r=q1q24πε0rU = k\frac{q_1 q_2}{r} = \frac{q_1 q_2}{4\pi\varepsilon_0 r}
Potential energy of the pair. U > 0 for like charges (energy needed to bring them together); U < 0 for unlike charges.
Electric Field from Potential
E=−∇V=−(∂V∂xx^+∂V∂yy^+∂V∂zz^)\mathbf{E} = -\nabla V = -\left(\frac{\partial V}{\partial x}\hat{x} + \frac{\partial V}{\partial y}\hat{y} + \frac{\partial V}{\partial z}\hat{z}\right)
E points from high potential to low potential, in the direction of steepest descent of V. Equipotential surfaces are always perpendicular to E.
Potential from a Continuous Distribution
V=k∫dqrV = k\int\frac{dq}{r}
Integrate the contribution kدق/r from each infinitesimal charge element dq at distance r from the field point. Scalar integral; easier than finding E directly.
Relation Between E and V (1D)
Ex=−dVdx,VB−VA=−∫ABE⋅dlE_x = -\frac{dV}{dx}, \qquad V_B - V_A = -\int_A^B \mathbf{E}\cdot d\mathbf{l}
E is the negative gradient of V. The line integral of E from A to B gives the potential drop.

Capacitance & Energy Storage

Capacitance Definition
C=QV,[C]=farads (F)=C/VC = \frac{Q}{V}, \qquad [C] = \text{farads (F)} = \text{C/V}
Capacitance is the charge stored per unit potential difference. Depends only on geometry and the material between the plates.
Parallel-Plate Capacitor
C=ε0Ad(vacuum),C=κε0Ad(dielectric κ)C = \frac{\varepsilon_0 A}{d} \quad (\text{vacuum}), \qquad C = \frac{\kappa\varepsilon_0 A}{d} \quad (\text{dielectric } \kappa)
A and d are the plate area and separation. Inserting a dielectric of constant κ increases capacitance by a factor of κ.
Capacitors in Series
1Ceq=1C1+1C2+⋯\frac{1}{C_{\rm eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots
In series, each capacitor stores the same charge Q. The voltage divides; the equivalent capacitance is less than any individual capacitor.
Capacitors in Parallel
Ceq=C1+C2+⋯C_{\rm eq} = C_1 + C_2 + \cdots
In parallel, each capacitor shares the same voltage. Charges add; equivalent capacitance is the sum of all capacitances.
Energy Stored in a Capacitor
U=Q22C=12CV2=12QVU = \frac{Q^2}{2C} = \frac{1}{2}CV^2 = \frac{1}{2}QV
Energy stored in the electric field between the plates. All three forms are equivalent via C = Q/V.
Electric Energy Density
uE=12ε0E2(vacuum),uE=12κε0E2(dielectric)u_E = \frac{1}{2}\varepsilon_0 E^2 \quad (\text{vacuum}), \qquad u_E = \frac{1}{2}\kappa\varepsilon_0 E^2 \quad (\text{dielectric})
Energy stored per unit volume in an electric field. Total energy U = ∫u_E dV. Valid for any electric field, not just capacitors.

Current, Resistance & Power

Electric Current
I=dQdt,[I]=amperes (A)=C/sI = \frac{dQ}{dt}, \qquad [I] = \text{amperes (A)} = \text{C/s}
Conventional current direction is the direction positive charges would flow. For electrons, current is opposite to drift velocity.
Current Density & Drift Velocity
I=nqvdA,J=IA=nqvdI = nqv_d A, \qquad J = \frac{I}{A} = nqv_d
n = number density of charge carriers, q = carrier charge, v_d = drift velocity, A = cross-sectional area. Drift speed is typically ~mm/s despite fast electrical signals.
Ohm's Law
V=IR,J=σE=EρV = IR, \qquad J = \sigma E = \frac{E}{\rho}
Macroscopic (V = IR) and microscopic (J = σE) forms. Ohm's law is a material property, not a fundamental law; it holds for many conductors over a wide range.
Resistance
R=ρLA,ρ=ρ0[1+α(T−T0)]R = \frac{\rho L}{A}, \qquad \rho = \rho_0[1 + \alpha(T - T_0)]
R depends on resistivity ρ (a material property), length L, and cross-sectional area A. Resistivity increases with temperature for most metals (α > 0).
Electric Power
P=IV=I2R=V2RP = IV = I^2 R = \frac{V^2}{R}
Power dissipated (as heat) in a resistor, or delivered by a source. All three forms follow from P = IV and Ohm's law.

DC Circuits

Kirchhoff's Junction Rule (KCL)
∑Iin=∑Iout⇔∑jIj=0 at a node\sum I_{\rm in} = \sum I_{\rm out} \quad \Leftrightarrow \quad \sum_j I_j = 0 \text{ at a node}
Conservation of charge: the sum of currents entering a junction equals the sum leaving. Apply at each node in the circuit.
Kirchhoff's Loop Rule (KVL)
∑loopΔV=0\sum_{\rm loop} \Delta V = 0
Conservation of energy: the sum of all potential rises and drops around any closed loop is zero. Cross a resistor in the current direction → drop −IR; a battery from − to + → rise +ε.
Resistors in Series & Parallel
Rseries=R1+R2+⋯1Rparallel=1R1+1R2+⋯R_{\rm series} = R_1 + R_2 + \cdots \qquad \frac{1}{R_{\rm parallel}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots
Series: same current, voltages add. Parallel: same voltage, currents add. For two resistors in parallel: R_eq = R₁R₂/(R₁ + R₂).
EMF and Terminal Voltage
Vterminal=ε−Ir,Pdelivered=εI−I2rV_{\rm terminal} = \varepsilon - Ir, \qquad P_{\rm delivered} = \varepsilon I - I^2 r
A real battery has internal resistance r. Terminal voltage drops below the EMF ε when current I flows. The I²r term is the power dissipated internally.
RC Circuit — Charging
q(t)=Cε(1−e−t/τ),I(t)=εRe−t/τ,τ=RCq(t) = C\varepsilon\bigl(1 - e^{-t/\tau}\bigr), \qquad I(t) = \frac{\varepsilon}{R}e^{-t/\tau}, \qquad \tau = RC
Capacitor charges exponentially toward Q_f = Cε with time constant τ = RC. At t = τ, the capacitor has reached 63% of its final charge.
RC Circuit — Discharging
q(t)=Q0 e−t/τ,I(t)=Q0RCe−t/τ,τ=RCq(t) = Q_0\,e^{-t/\tau}, \qquad I(t) = \frac{Q_0}{RC}e^{-t/\tau}, \qquad \tau = RC
Capacitor discharges exponentially through the resistor. The charge (and voltage) fall to 1/e ≈ 37% of Q₀ after one time constant τ.

Magnetic Force & Field

Magnetic Force on a Moving Charge
F=qv×B,∣F∣=qvBsin⁡θ\mathbf{F} = q\mathbf{v}\times\mathbf{B}, \qquad |F| = qvB\sin\theta
The magnetic force is always perpendicular to both v and B, so it does no work and cannot change kinetic energy. The direction follows the right-hand rule.
Lorentz Force (combined E and B)
F=q(E+v×B)\mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B})
Total electromagnetic force on a charge q moving with velocity v in fields E and B.
Cyclotron Motion
r=mvqB,ωc=qBm,T=2πmqBr = \frac{mv}{qB}, \qquad \omega_c = \frac{qB}{m}, \qquad T = \frac{2\pi m}{qB}
A charge moving perpendicular to a uniform B traces a circle. Radius r depends on momentum; the cyclotron frequency ω_c is independent of speed.
Magnetic Force on a Current
F=IL×B,∣F∣=BILsin⁡θ\mathbf{F} = I\mathbf{L}\times\mathbf{B}, \qquad |F| = BIL\sin\theta
Force on a straight current-carrying conductor of length L in field B. L is in the direction of conventional current flow.
Magnetic Dipole Moment
μ=IA=NIAn^,τ=μ×B,U=−μ⋅B\boldsymbol{\mu} = I\mathbf{A} = NIA\hat{n}, \qquad \boldsymbol{\tau} = \boldsymbol{\mu}\times\mathbf{B}, \qquad U = -\boldsymbol{\mu}\cdot\mathbf{B}
A current loop of area A and N turns acts as a magnetic dipole. Torque tends to align μ with B; minimum energy when parallel.
Magnetic Flux
ΦB=∫SB⋅dA=BAcos⁡θ(uniform B)\Phi_B = \int_S \mathbf{B}\cdot d\mathbf{A} = BA\cos\theta \quad (\text{uniform }\mathbf{B})
Magnetic flux through a surface S. Units: weber (Wb = T·m²). Used in Faraday's law and Gauss's law for magnetism.

Sources of Magnetic Field

Biot-Savart Law
dB=μ04πI dl×r^r2,μ0=4π×10−7 T⋅m/Ad\mathbf{B} = \frac{\mu_0}{4\pi}\frac{I\,d\mathbf{l}\times\hat{r}}{r^2}, \qquad \mu_0 = 4\pi\times10^{-7}\text{ T·m/A}
Magnetic field contribution from current element I dl at distance r. μ₀ is the permeability of free space. Integrate over the full wire to find B.
Field of an Infinite Straight Wire
B=μ0I2πrB = \frac{\mu_0 I}{2\pi r}
Field at perpendicular distance r from a long straight wire carrying current I. Field lines are concentric circles; direction given by right-hand rule.
Ampère's Law
∮CB⋅dl=μ0Ienc\oint_C \mathbf{B}\cdot d\mathbf{l} = \mu_0 I_{\rm enc}
The line integral of B around any closed Amperian loop equals μ₀ times the current threading the loop. Most useful for high-symmetry geometries.
Field Inside a Solenoid
B=μ0nI=μ0NILB = \mu_0 n I = \frac{\mu_0 N I}{L}
Uniform field inside an ideal (infinite) solenoid with n turns per unit length (n = N/L). Field is zero outside. Excellent model for electromagnets.
Field Inside a Toroid
B=μ0NI2πrB = \frac{\mu_0 N I}{2\pi r}
Field inside a toroidal solenoid with N turns at radius r from the center. Field is zero outside the toroid.
Force Between Parallel Wires
FL=μ0I1I22πd\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}
Force per unit length between two long parallel wires carrying currents I₁ and I₂ separated by distance d. Parallel currents attract; antiparallel repel. Defined the ampere historically.

Electromagnetic Induction

Faraday's Law
ε=−dΦBdt=−NdΦBdt(N turns)\varepsilon = -\frac{d\Phi_B}{dt} = -N\frac{d\Phi_B}{dt} \quad (N \text{ turns})
A changing magnetic flux through a circuit induces an EMF. The negative sign is Lenz's law: the induced effects oppose the cause.
Lenz's Law
The induced current creates a magnetic field that opposes the change in flux.\text{The induced current creates a magnetic field that opposes the change in flux.}
A consequence of energy conservation. If the flux through a loop is increasing, the induced current creates a field opposing that increase.
Motional EMF
ε=BLvsin⁡θ,ε=∫(v×B)⋅dl\varepsilon = BLv\sin\theta, \qquad \varepsilon = \int(\mathbf{v}\times\mathbf{B})\cdot d\mathbf{l}
EMF induced in a conductor of length L moving with velocity v at angle θ to field B. Charges in the moving conductor experience a v × B force.
Faraday's Law (integral form)
∮CE⋅dl=−ddt∫SB⋅dA\oint_C \mathbf{E}\cdot d\mathbf{l} = -\frac{d}{dt}\int_S \mathbf{B}\cdot d\mathbf{A}
The line integral of E around a closed loop equals the negative rate of change of magnetic flux through any surface bounded by that loop.
Generator EMF
ε(t)=NBAωsin⁡(ωt)=ε0sin⁡(ωt)\varepsilon(t) = NBA\omega\sin(\omega t) = \varepsilon_0\sin(\omega t)
EMF from a rectangular coil of N turns and area A rotating at angular velocity ω in uniform field B. The peak EMF is ε₀ = NBAω.

Inductance & AC Circuits

Self-Inductance
εL=−LdIdt,L=NΦBI,[L]=henry (H)\varepsilon_L = -L\frac{dI}{dt}, \qquad L = \frac{N\Phi_B}{I}, \qquad [L] = \text{henry (H)}
An inductor opposes changes in current. L is the self-inductance; N turns × flux per turn / current.
Solenoid Inductance
L=μ0n2V=μ0N2AℓL = \mu_0 n^2 V = \frac{\mu_0 N^2 A}{\ell}
Inductance of a solenoid with N turns, cross-sectional area A, and length ℓ. V = Aℓ is the solenoid volume; n = N/ℓ.
Energy Stored in an Inductor
UL=12LI2,uB=B22μ0U_L = \frac{1}{2}LI^2, \qquad u_B = \frac{B^2}{2\mu_0}
Energy stored in the magnetic field of an inductor carrying current I. The energy density u_B is valid for any magnetic field.
LC Oscillation
ω0=1LC,T=2πLC\omega_0 = \frac{1}{\sqrt{LC}}, \qquad T = 2\pi\sqrt{LC}
Natural frequency of an ideal LC circuit. Energy oscillates between the electric field (capacitor) and magnetic field (inductor), analogous to a mass-spring system.
Reactances & Impedance (RLC Series)
XC=1ωC,XL=ωL,Z=R2+(XL−XC)2X_C = \frac{1}{\omega C}, \quad X_L = \omega L, \quad Z = \sqrt{R^2 + (X_L - X_C)^2}
Capacitive reactance X_C decreases with frequency; inductive reactance X_L increases. Impedance Z is the AC generalization of resistance.
AC Resonance
ωr=1LC,Zmin=R at resonance,Imax=V0R\omega_r = \frac{1}{\sqrt{LC}}, \qquad Z_{\rm min} = R \text{ at resonance}, \qquad I_{\rm max} = \frac{V_0}{R}
At resonance X_L = X_C, impedance is purely resistive and current is maximum. The quality factor Q = ω_r L/R = (1/R)√(L/C) measures the sharpness of the resonance.
Average Power & RMS Values
Irms=I02,Vrms=V02,Pavg=IrmsVrmscos⁡ϕI_{\rm rms} = \frac{I_0}{\sqrt{2}}, \quad V_{\rm rms} = \frac{V_0}{\sqrt{2}}, \quad P_{\rm avg} = I_{\rm rms}V_{\rm rms}\cos\phi
RMS (root-mean-square) values are the DC equivalents for power. cos φ = R/Z is the power factor; φ is the phase angle between current and voltage.

Maxwell's Equations

Gauss's Law for Electricity (integral)
∮SE⋅dA=Qencε0⇔∇⋅E=ρε0\oint_S \mathbf{E}\cdot d\mathbf{A} = \frac{Q_{\rm enc}}{\varepsilon_0} \quad\Leftrightarrow\quad \nabla\cdot\mathbf{E} = \frac{\rho}{\varepsilon_0}
Electric field lines originate on positive charges and end on negative charges. The enclosed charge is the source of the electric flux.
Gauss's Law for Magnetism (integral)
∮SB⋅dA=0⇔∇⋅B=0\oint_S \mathbf{B}\cdot d\mathbf{A} = 0 \quad\Leftrightarrow\quad \nabla\cdot\mathbf{B} = 0
No magnetic monopoles exist. Magnetic field lines always form closed loops; every line that enters a closed surface must also exit.
Faraday's Law (integral)
∮CE⋅dl=−dΦBdt⇔∇×E=−∂B∂t\oint_C \mathbf{E}\cdot d\mathbf{l} = -\frac{d\Phi_B}{dt} \quad\Leftrightarrow\quad \nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}
A changing magnetic field induces a circulating electric field. This is the foundation of electric generators and transformers.
Ampère-Maxwell Law (integral)
∮CB⋅dl=μ0 ⁣(Ienc+ε0dΦEdt)⇔∇×B=μ0J+μ0ε0∂E∂t\oint_C \mathbf{B}\cdot d\mathbf{l} = \mu_0\!\left(I_{\rm enc} + \varepsilon_0\frac{d\Phi_E}{dt}\right) \quad\Leftrightarrow\quad \nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}
Maxwell's key addition: the displacement current ε₀ dΦ_E/dt. A changing electric field generates a magnetic field, even without real current. This term gives self-sustaining electromagnetic waves.
Speed of Light (from Maxwell)
c=1μ0ε0≈3.00×108 m/sc = \frac{1}{\sqrt{\mu_0\varepsilon_0}} \approx 3.00\times10^8\text{ m/s}
Maxwell derived that electromagnetic waves propagate at this speed — matching the known speed of light. This identified light as an electromagnetic wave.
Displacement Current
Id=ε0dΦEdtI_d = \varepsilon_0\frac{d\Phi_E}{dt}
Not a real current but a changing electric flux. It appears in the Ampère-Maxwell law and ensures current is continuous (e.g., inside a charging capacitor).

Electromagnetic Waves

Wave Equation for E and B
∇2E=μ0ε0∂2E∂t2,∇2B=μ0ε0∂2B∂t2\nabla^2\mathbf{E} = \mu_0\varepsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2}, \qquad \nabla^2\mathbf{B} = \mu_0\varepsilon_0\frac{\partial^2\mathbf{B}}{\partial t^2}
E and B each satisfy the wave equation with wave speed c = 1/√(μ₀ε₀). Derived from Maxwell's equations in free space.
Plane Wave (traveling in +x)
E=E0cos⁡(kx−ωt),B=B0cos⁡(kx−ωt),E0B0=cE = E_0\cos(kx - \omega t), \quad B = B_0\cos(kx - \omega t), \quad \frac{E_0}{B_0} = c
E and B are in phase, mutually perpendicular, and both perpendicular to the propagation direction (transverse wave). E₀/B₀ = c always.
Intensity (Poynting Vector)
S=1μ0E×B,I=⟨S⟩=E022μ0c=cB022μ0=cE02ε02\mathbf{S} = \frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}, \qquad I = \langle S\rangle = \frac{E_0^2}{2\mu_0 c} = \frac{cB_0^2}{2\mu_0} = \frac{cE_0^2\varepsilon_0}{2}
The Poynting vector S = (E × B)/μ₀ gives the energy flux (W/m²). The time-averaged intensity I is S averaged over a full cycle.
Radiation Pressure
Prad=Ic(absorbed),Prad=2Ic(reflected)P_{\rm rad} = \frac{I}{c} \quad (\text{absorbed}), \qquad P_{\rm rad} = \frac{2I}{c} \quad (\text{reflected})
Electromagnetic waves carry momentum. The radiation pressure on a perfectly absorbing surface is I/c; on a perfectly reflecting surface, 2I/c.

Geometric Optics

Law of Reflection
θr=θi\theta_r = \theta_i
The angle of reflection equals the angle of incidence, both measured from the surface normal. Applies to any smooth (specular) surface.
Index of Refraction
n=cv,nair≈1.000,nglass≈1.5,nwater≈1.33n = \frac{c}{v}, \qquad n_{\rm air} \approx 1.000, \quad n_{\rm glass} \approx 1.5, \quad n_{\rm water} \approx 1.33
Ratio of the speed of light in vacuum to the speed in the medium. n ≥ 1 always; denser media have higher n and slower light.
Snell's Law (Refraction)
n1sin⁡θ1=n2sin⁡θ2n_1\sin\theta_1 = n_2\sin\theta_2
Light bends toward the normal when entering a denser medium (larger n), and away from the normal when entering a less-dense medium.
Total Internal Reflection
sin⁡θc=n2n1(n1>n2)\sin\theta_c = \frac{n_2}{n_1} \quad (n_1 > n_2)
Above the critical angle θ_c, all light is reflected back into the denser medium. Basis of fiber optics and prismatic reflectors.
Mirror & Thin-Lens Equation
1do+1di=1f,fmirror=R2\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}, \qquad f_{\rm mirror} = \frac{R}{2}
d_o = object distance, d_i = image distance, f = focal length. Sign convention: real is positive for lenses; for mirrors, real object/image distances are positive on the reflecting side.
Lateral Magnification
m=−dido=hihom = -\frac{d_i}{d_o} = \frac{h_i}{h_o}
|m| > 1: magnified; |m| < 1: reduced. m > 0: upright (virtual) image; m < 0: inverted (real) image. Same formula for mirrors and thin lenses.
Lensmaker's Equation
1f=(n−1) ⁣(1R1−1R2)\frac{1}{f} = (n-1)\!\left(\frac{1}{R_1} - \frac{1}{R_2}\right)
Relates focal length to the radii of curvature R₁, R₂ of the two lens surfaces and the index n. Sign convention: R > 0 if center of curvature is to the right.

Wave Optics: Interference

Double-Slit Bright Fringes
dsin⁡θ=mλ,m=0,±1,±2,…d\sin\theta = m\lambda, \quad m = 0,\pm1,\pm2,\ldots
Constructive interference (bright fringes) when the path difference d sinθ is an integer number of wavelengths. d = slit separation.
Double-Slit Dark Fringes
dsin⁡θ=(m+12)λ,m=0,±1,±2,…d\sin\theta = \left(m + \frac{1}{2}\right)\lambda, \quad m = 0,\pm1,\pm2,\ldots
Destructive interference (dark fringes) when path difference is a half-integer multiple of λ.
Fringe Spacing
Δy=λLd\Delta y = \frac{\lambda L}{d}
Distance between adjacent bright (or dark) fringes on a screen at distance L. Valid for small angles (L ≫ d).
Thin-Film Interference — Phase Inversion
Phase inversion occurs when light reflects off a medium with nmedium>nincident\text{Phase inversion occurs when light reflects off a medium with } n_{\rm medium} > n_{\rm incident}
Reflection from a slower medium (higher n) inverts phase by π (equivalent to half-wavelength path shift). Reflection from a faster medium has no phase inversion.
Thin-Film: Constructive Interference
2nt={mλ0 or 2 inversions(m+12)λ1 inversion,m=0,1,2,…2nt = \begin{cases}m\lambda & \text{0 or 2 inversions}\\(m+\tfrac{1}{2})\lambda & \text{1 inversion}\end{cases}, \quad m = 0,1,2,\ldots
The round-trip optical path in the film is 2nt (n = film index, t = thickness). Phase inversions at each boundary must be accounted for.
Condition for Destructive Interference (thin film)
2nt={(m+12)λ0 or 2 inversionsmλ1 inversion2nt = \begin{cases}(m+\tfrac{1}{2})\lambda & \text{0 or 2 inversions}\\m\lambda & \text{1 inversion}\end{cases}
Swap the constructive/destructive conditions from the bright-fringe case depending on the number of phase inversions at the interfaces.

Wave Optics: Diffraction

Single-Slit Dark Fringes
asin⁡θ=mλ,m=±1,±2,…a\sin\theta = m\lambda, \quad m = \pm1,\pm2,\ldots
Minima of single-slit diffraction pattern for slit width a. Note m ≠ 0 here; the central maximum is between the m = ±1 minima.
Single-Slit Intensity Pattern
I=I0(sin⁡αα) ⁣2,α=πasin⁡θλI = I_0\left(\frac{\sin\alpha}{\alpha}\right)^{\!2}, \qquad \alpha = \frac{\pi a\sin\theta}{\lambda}
Full intensity distribution for single-slit diffraction. Central maximum has width 2λ/a; subsidiary maxima are much weaker.
Diffraction Grating — Principal Maxima
dsin⁡θ=mλ,m=0,±1,±2,…d\sin\theta = m\lambda, \quad m = 0,\pm1,\pm2,\ldots
Same form as double-slit, but here d is the grating spacing and N slits create extremely sharp, bright maxima. Used in spectrometers.
Resolving Power of a Diffraction Grating
R=λΔλmin=mNR = \frac{\lambda}{\Delta\lambda_{\rm min}} = mN
A grating with N slits in diffraction order m can just resolve two wavelengths differing by Δλ_min. More slits and higher orders give finer wavelength discrimination.
Rayleigh Criterion (circular aperture)
θmin=1.22λD\theta_{\rm min} = 1.22\frac{\lambda}{D}
Two point sources are just resolvable when the central maximum of one falls on the first minimum of the other. D = aperture diameter. Sets the angular resolution of telescopes, microscopes, and the eye.
X-Ray Bragg Diffraction
2dsin⁡θ=mλ,m=1,2,3,…2d\sin\theta = m\lambda, \quad m = 1,2,3,\ldots
Constructive interference of X-rays reflected from parallel crystal planes separated by lattice spacing d. Used in X-ray crystallography to determine atomic structure.