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Waves, Optics & EM Formulas

All wave, optics, and electromagnetism equations with definitions and units.

Wave Properties

Wave Equation
v = fλ
v=speed (m/s), f=frequency (Hz), λ=wavelength (m)
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Period
T = 1/f
T=period (seconds), f=frequency (Hz)
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Wave Number
k = 2π/λ
k in rad/m; angular frequency ω=2πf (rad/s)
Wave Function
y=A sin(kx−ωt)
Transverse wave traveling in +x direction
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Speed in String
v = √(T/μ)
T=tension (N), μ=mass per unit length (kg/m)
Sound Speed (gas)
v = √(γRT/M)
γ=ratio Cp/Cv, R=8.314, T in K, M=molar mass
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Sound and Acoustics

Sound Intensity
I = P/A
I in W/m², P=power (W), A=area (m²)
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Sound Level
β = 10log(I/I₀)
β in dB, I₀=10⁻¹² W/m²
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Doppler Effect
f'=f(v±v_o)/(v∓v_s)
+ for approaching, − for receding
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Beat Frequency
f_beat = |f₁ − f₂|
Heard as periodic loudness variation
Open Pipe Harmonics
f_n = nv/2L
n=1,2,3...; L=pipe length
Closed Pipe Harmonics
f_n=(2n−1)v/4L
n=1,2,3...; closed at one end

Optics

Law of Reflection
θ_i = θ_r
Both angles measured from the normal to the surface
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Snell's Law
n₁sinθ₁ = n₂sinθ₂
n=refractive index; n=c/v
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Refractive Index
n = c/v
c=3×10⁸ m/s; n_glass≈1.5; n_water≈1.33
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Critical Angle
sinθ_c = n₂/n₁
For total internal reflection; n₁>n₂ required
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Mirror/Lens Equation
1/f = 1/d_o + 1/d_i
Magnification m = −d_i/d_o

Electrostatics

Coulomb's Law
F = kq₁q₂/r²
k=8.99×10⁹ N·m²/C²
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Electric Field
E = F/q = kQ/r²
E in N/C or V/m
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Electric Potential
V = kQ/r
V in Volts (J/C)
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Potential Energy
U = qV = kq₁q₂/r
U in Joules
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Gauss's Law
∮E·dA = Q_enc/ε₀
ε₀=8.854×10⁻¹² F/m
Capacitance
C = Q/V = ε₀A/d
C in Farads (F); parallel-plate capacitor formula
Capacitor Energy
U = ½CV² = Q²/2C
Energy stored in electric field

Electric Circuits

Ohm's Law
V = IR
V=volts, I=amperes, R=ohms
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Resistivity
R = ρL/A
ρ in Ω·m; copper ρ≈1.7×10⁻⁸
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Electric Power
P = IV = I²R = V²/R
P in watts; I²R is Joule heating
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Series Resistance
R_T = R₁+R₂+...
Same current through all
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Parallel Resistance
1/R_T=1/R₁+1/R₂
Same voltage across all
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Electric Current
I = dQ/dt
I in amperes (C/s)

Modern Physics

Photon Energy
E = hf = hc/λ
h=6.626×10⁻³⁴ J·s
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Photoelectric Effect
KE_max = hf − φ
φ=work function of metal (eV)
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de Broglie
λ = h/p = h/(mv)
Matter wavelength
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Mass-Energy
E = mc²
Rest energy; c=3×10⁸ m/s
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Time Dilation
Δt' = γΔt₀
γ=1/√(1−v²/c²)
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Uncertainty Principle
ΔxΔp ≥ ℏ/2
ℏ=1.055×10⁻³⁴ J·s
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