Physics formulas: Electrodynamics

Quick reference for school physics (electrodynamics). Browse by topic or search by name.

Electrostatics

Coulomb's law

F=k∣q1q2∣r2F = \frac{k|q_1 q_2|}{r^2}
Symbols
  • FF — magnitude of the force between point charges, N (newton)
  • kk — Coulomb constant, 9×10⁹ N·m²/C²
  • q1,q2q_1, q_2 — magnitudes of the charges, C (coulomb)
  • rr — distance between the charges, m (meter)
k=9⋅109 N⋅m2/C2k = 9 \cdot 10^9\,\text{N}\cdot\text{m}^2/\text{C}^2 is Coulomb's constant.
Electrostatics

Electric field strength

E=FqE = \frac{F}{q}
Symbols
  • EE — electric field intensity, V/m
  • FF — force acting on the test charge, N (newton)
  • qq — magnitude of the test charge, C (coulomb)
Electrostatics

Electric field of a point charge

E=k∣q∣r2E = \frac{k|q|}{r^2}
Symbols
  • EE — electric field of a point charge, V/m
  • kk — Coulomb constant, 9×10⁹ N·m²/C²
  • qq — magnitude of the charge, C (coulomb)
  • rr — distance from the charge to the field point, m (meter)
Electrostatics

Electric field of a charged sphere

E=k∣q∣r2E = \frac{k|q|}{r^2}
E=0E = 0
Symbols
  • EE — electric field intensity, V/m
  • kk — Coulomb constant, 9×10⁹ N·m²/C²
  • qq — total charge of the sphere, C (coulomb)
  • rr — distance from the center of the sphere (outside), m (meter)
  • RR — radius of the charged sphere, m (meter)
Outside the sphere (r>Rr > R) — first formula; inside a conductor (r≤Rr \leq R) — E=0E = 0.
Electrostatics

Principle of superposition of electric fields

E=E1+E2+E3+…E = E_1 + E_2 + E_3 + \ldots
Symbols
  • EE — resultant electric field, V/m
  • E1,E2,…E_1, E_2, \ldots — field intensities of individual charges, V/m
Electrostatics

Electric potential

φ=Wq\varphi = \frac{W}{q}
Symbols
  • φ\varphi — electric potential at a field point, V (volt)
  • WW — work of electric forces when moving the charge, J (joule)
  • qq — charge being moved, C (coulomb)
Electrostatics

Potential of a point charge

φ=kqr\varphi = \frac{kq}{r}
Symbols
  • φ\varphi — potential of a point charge field, V (volt)
  • kk — Coulomb constant, 9×10⁹ N·m²/C²
  • qq — magnitude of the charge, C (coulomb)
  • rr — distance from the charge to the field point, m (meter)
Electrostatics

Potential of a charged sphere

φ=kqr\varphi = \frac{kq}{r}
φ=kqR\varphi = \frac{kq}{R}
Symbols
  • φ\varphi — electric potential, V (volt)
  • kk — Coulomb constant, 9×10⁹ N·m²/C²
  • qq — total charge of the sphere, C (coulomb)
  • rr — distance from the center of the sphere (outside), m (meter)
  • RR — radius of the charged sphere, m (meter)
Outside (r>Rr > R) — φ=kq/r\varphi = kq/r; on the surface and inside (r≤Rr \leq R) — φ=kq/R\varphi = kq/R.
Electrostatics

Potential in a uniform electric field

φ=Ed\varphi = Ed
Symbols
  • φ\varphi — potential difference between the points, V (volt)
  • EE — intensity of the uniform field, V/m
  • dd — distance along the field lines, m (meter)
Electrostatics

Potential of a system of point charges

φ=φ1+φ2+…\varphi = \varphi_1 + \varphi_2 + \ldots
Symbols
  • φ\varphi — total potential at the point, V (volt)
  • φ1,φ2,…\varphi_1, \varphi_2, \ldots — potentials of individual charges, V (volt)
Electrostatics

Work moving a charge in an electric field

W=qUW = qU
Symbols
  • WW — work done by the electric field, J (joule)
  • qq — charge being moved, C (coulomb)
  • UU — potential difference (voltage), V (volt)
Electrostatics

Relation between field strength and voltage

E=UdE = \frac{U}{d}
Symbols
  • EE — intensity of a uniform electric field, V/m
  • UU — voltage between the plates, V (volt)
  • dd — distance between the plates, m (meter)
Electrostatics

Potential energy of two point charges

W=kq1q2rW = \frac{kq_1 q_2}{r}
Symbols
  • WW — potential energy of interaction of two charges, J (joule)
  • kk — Coulomb constant, 9×10⁹ N·m²/C²
  • q1,q2q_1, q_2 — magnitudes of the charges, C (coulomb)
  • rr — distance between the charges, m (meter)
Electrostatics

Capacitance

C=qUC = \frac{q}{U}
Symbols
  • CC — capacitance of the capacitor, F (farad)
  • qq — charge on the plates, C (coulomb)
  • UU — voltage across the capacitor, V (volt)
Electrostatics

Capacitance of a parallel-plate capacitor

C=εε0SdC = \frac{\varepsilon\varepsilon_0 S}{d}
Symbols
  • CC — capacitance of a parallel-plate capacitor, F (farad)
  • ε\varepsilon — relative permittivity of the medium, dimensionless quantity
  • ε0\varepsilon_0 — electric constant (vacuum permittivity), 8.85×10⁻¹² F/m
  • SS — area of the plates, m² (square meter)
  • dd — distance between the plates, m (meter)
Electrostatics

Capacitors in parallel

C=C1+C2+…C = C_1 + C_2 + \ldots
Symbols
  • CC — total capacitance in parallel connection, F (farad)
  • C1,C2,…C_1, C_2, \ldots — capacitances of individual capacitors, F (farad)
Electrostatics

Capacitors in series

1C=1C1+1C2+…\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots
Symbols
  • CC — total capacitance in series connection, F (farad)
  • C1,C2,…C_1, C_2, \ldots — capacitances of individual capacitors, F (farad)
Electrostatics

Energy of a charged capacitor

W=CU22W = \frac{CU^2}{2}
Symbols
  • WW — energy stored in the capacitor electric field, J (joule)
  • CC — capacitance of the capacitor, F (farad)
  • UU — voltage across the capacitor, V (volt)
Also W=q22CW = \frac{q^2}{2C} and W=qU2W = \frac{qU}{2}.
Electrostatics

Surface charge density

σ=qS\sigma = \frac{q}{S}
Symbols
  • σ\sigma — surface charge density, C/m²
  • qq — charge on the surface, C (coulomb)
  • SS — area of the surface, m² (square meter)
Direct current

Electric current

I=qtI = \frac{q}{t}
Symbols
  • II — electric current, A (ampere)
  • qq — electric charge passing through the conductor cross-section, C (coulomb)
  • tt — time during which the charge passes, s (second)
Direct current

Current (drift model)

I=neSvI = neSv
Symbols
  • II — electric current, A (ampere)
  • nn — concentration of charge carriers, m⁻³
  • ee — charge of one carrier (electron), 1.6×10⁻¹⁹ C
  • SS — cross-sectional area of the conductor, m² (square meter)
  • vv — drift speed of the charge carriers, m/s
Direct current

Current density

j=IS=nevj = \frac{I}{S} = nev
Symbols
  • jj — electric current density, A/m²
  • II — electric current, A (ampere)
  • SS — cross-sectional area, m² (square meter)
  • nn — concentration of charge carriers, m⁻³
  • ee — charge of one carrier, 1.6×10⁻¹⁹ C
  • vv — drift speed of the carriers, m/s
Direct current

Ohm's law

I=URI = \frac{U}{R}
Symbols
  • II — current in the circuit section, A (ampere)
  • UU — voltage across the section, V (volt)
  • RR — electrical resistance of the section, Ω (ohm)
Direct current

Resistance of a conductor

R=ρlSR = \frac{\rho l}{S}
Symbols
  • RR — resistance of the conductor, Ω (ohm)
  • ρ\rho — resistivity of the material, Ω·m
  • ll — length of the conductor, m (meter)
  • SS — cross-sectional area, m² (square meter)
Direct current

Temperature dependence of resistance

R=R0(1+α(t−t0))R = R_0(1 + \alpha(t - t_0))
Symbols
  • RR — resistance at temperature t, Ω (ohm)
  • R0R_0 — resistance at the reference temperature t_0, Ω (ohm)
  • α\alpha — temperature coefficient of resistance, °C⁻¹
  • tt — current temperature, °C (degree Celsius)
  • t0t_0 — reference temperature, °C (degree Celsius)
Direct current

Resistors in series

R=R1+R2+…R = R_1 + R_2 + \ldots
Symbols
  • RR — total resistance in series connection, Ω (ohm)
  • R1,R2,…R_1, R_2, \ldots — resistances of individual resistors, Ω (ohm)
Direct current

Resistors in parallel

1R=1R1+1R2+…\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots
Symbols
  • RR — total resistance in parallel connection, Ω (ohm)
  • R1,R2,…R_1, R_2, \ldots — resistances of individual resistors, Ω (ohm)
Direct current

Work of electric current

W=UItW = UIt
Symbols
  • WW — work done by the electric current, J (joule)
  • UU — voltage across the circuit section, V (volt)
  • II — electric current, A (ampere)
  • tt — time during which the current flows, s (second)
Direct current

Electric power

P=UIP = UI
Symbols
  • PP — electric power, W (watt)
  • UU — voltage, V (volt)
  • II — electric current, A (ampere)
Direct current

Joule–Lenz law

Q=I2RtQ = I^2 R t
Symbols
  • QQ — heat released in the conductor, J (joule)
  • II — electric current, A (ampere)
  • RR — resistance of the conductor, Ω (ohm)
  • tt — time during which the current flows, s (second)
Direct current

Electromotive force (EMF)

E=Wextq\mathcal{E} = \frac{W_{\text{ext}}}{q}
Symbols
  • E\mathcal{E} — electromotive force (EMF), V (volt)
  • WextW_{\text{ext}} — work of external forces moving charge around a closed loop, J (joule)
  • qq — charge being moved, C (coulomb)
Direct current

Ohm's law for a complete circuit

I=ER+rI = \frac{\mathcal{E}}{R + r}
Symbols
  • II — current in the complete circuit, A (ampere)
  • E\mathcal{E} — EMF of the source, V (volt)
  • RR — external resistance of the circuit, Ω (ohm)
  • rr — internal resistance of the source, Ω (ohm)
Direct current

Current with sources connected in series

I=nER+nrI = \frac{n\mathcal{E}}{R + nr}
Symbols
  • II — current with sources connected in series, A (ampere)
  • nn — number of identical sources, dimensionless quantity
  • E\mathcal{E} — EMF of one source, V (volt)
  • RR — external resistance, Ω (ohm)
  • rr — internal resistance of one source, Ω (ohm)
Direct current

Current with sources connected in parallel

I=ER+r/nI = \frac{\mathcal{E}}{R + r/n}
Symbols
  • II — current with sources connected in parallel, A (ampere)
  • E\mathcal{E} — EMF of one source, V (volt)
  • RR — external resistance, Ω (ohm)
  • rr — internal resistance of one source, Ω (ohm)
  • nn — number of identical sources, dimensionless quantity
Direct current

Faraday's law of electrolysis

m=kItm = kIt
Symbols
  • mm — mass of substance deposited on the electrode, kg (kilogram)
  • kk — electrochemical equivalent of the substance, kg/(A·s)
  • II — electric current, A (ampere)
  • tt — time of electrolysis, s (second)
Magnetic field of electric current

Magnitude of magnetic induction

B=FIlB = \frac{F}{Il}
Symbols
  • BB — magnitude of the magnetic induction vector, T (tesla)
  • FF — Ampère force on the conductor (perpendicular to B), N (newton)
  • II — current in the conductor, A (ampere)
  • ll — length of the conductor segment in the field, m (meter)
When force FF is perpendicular to B⃗\vec{B}.
Magnetic field of electric current

Ampère's force

F=BIℓsin⁡αF = BI\ell\sin\alpha
Symbols
  • FF — force on a current-carrying conductor in a magnetic field, N (newton)
  • BB — magnetic induction, T (tesla)
  • II — electric current, A (ampere)
  • ℓ\ell — length of the conductor in the field, m (meter)
  • α\alpha — angle between B and the current direction, rad
Magnetic field of electric current

Lorentz force

F=qvBsin⁡αF = qvB\sin\alpha
Symbols
  • FF — Lorentz force acting on a charged particle, N (newton)
  • qq — charge of the particle, C (coulomb)
  • vv — speed of the particle, m/s
  • BB — magnetic induction, T (tesla)
  • α\alpha — angle between v and B, rad
Magnetic field of electric current

Momentum of a charged particle in a magnetic field

p=qBRp = qBR
Symbols
  • pp — momentum of the charged particle in the magnetic field, kg·m/s
  • qq — charge of the particle, C (coulomb)
  • BB — magnetic induction, T (tesla)
  • RR — radius of the particle trajectory (circle), m (meter)
Magnetic field of electric current

Magnetic flux

Φ=BScos⁡α\Phi = BS\cos\alpha
Symbols
  • Φ\Phi — magnetic flux through a surface, Wb (weber)
  • BB — magnetic induction, T (tesla)
  • SS — area of the surface, m² (square meter)
  • α\alpha — angle between B and the surface normal, rad
Electromagnetic induction

Law of electromagnetic induction

E=−ΔΦΔt\mathcal{E} = -\frac{\Delta\Phi}{\Delta t}
Symbols
  • E\mathcal{E} — induced EMF, V (volt)
  • ΔΦ\Delta\Phi — change in magnetic flux, Wb (weber)
  • Δt\Delta t — time interval of the flux change, s (second)
Electromagnetic induction

Magnetic flux through a surface

Φ=BScos⁡α\Phi = BS\cos\alpha
Symbols
  • Φ\Phi — magnetic flux through a surface, Wb (weber)
  • BB — magnetic induction, T (tesla)
  • SS — area of the surface, m² (square meter)
  • α\alpha — angle between B and the surface normal, rad
Electromagnetic induction

Maximum EMF of a frame rotating in a magnetic field

Emax⁡=BSω\mathcal{E}_{\max} = BS\omega
Symbols
  • Emax⁡\mathcal{E}_{\max} — maximum induced EMF in a rotating frame, V (volt)
  • BB — magnetic induction, T (tesla)
  • SS — area of the frame, m² (square meter)
  • ω\omega — angular velocity of rotation of the frame, rad/s
Electromagnetic induction

Self-induction EMF

Es=−L ΔIΔt\mathcal{E}_s = -L\,\frac{\Delta I}{\Delta t}
Symbols
  • Es\mathcal{E}_s — self-induction EMF, V (volt)
  • LL — inductance of the coil, H (henry)
  • ΔI\Delta I — change in current, A (ampere)
  • Δt\Delta t — time interval of the current change, s (second)
Electromagnetic induction

Motional EMF in a moving conductor

E=Blvsin⁡α\mathcal{E} = Blv\sin\alpha
Symbols
  • E\mathcal{E} — motional EMF of a conductor in a magnetic field, V (volt)
  • BB — magnetic induction, T (tesla)
  • ll — length of the conductor, m (meter)
  • vv — speed of the conductor, m/s
  • α\alpha — angle between B and v, rad
Electromagnetic induction

Induced charge

q=ΔΦRq = \frac{\Delta\Phi}{R}
Symbols
  • qq — induced charge in the circuit, C (coulomb)
  • ΔΦ\Delta\Phi — change in magnetic flux, Wb (weber)
  • RR — resistance of the circuit, Ω (ohm)
Electromagnetic oscillations

Oscillating charge

q(t)=qmcos⁡ωtq(t) = q_m\cos\omega t
Symbols
  • q(t)q(t) — charge on the capacitor plates at time t, C (coulomb)
  • qmq_m — amplitude of charge oscillations, C (coulomb)
  • ω\omega — angular frequency of oscillations, rad/s
  • tt — time, s (second)
Electromagnetic oscillations

Oscillating voltage

u(t)=umcos⁡ωtu(t) = u_m\cos\omega t
Symbols
  • u(t)u(t) — voltage across the capacitor at time t, V (volt)
  • umu_m — amplitude of voltage oscillations, V (volt)
  • ω\omega — angular frequency, rad/s
  • tt — time, s (second)
Electromagnetic oscillations

Oscillating current

i(t)=imcos⁡(ωt+π2)i(t) = i_m\cos\left(\omega t + \frac{\pi}{2}\right)
Symbols
  • i(t)i(t) — current in the circuit at time t, A (ampere)
  • imi_m — amplitude of current oscillations, A (ampere)
  • ω\omega — angular frequency, rad/s
  • tt — time, s (second)
Electromagnetic oscillations

Maximum oscillation current

im=ωqmi_m = \omega q_m
Symbols
  • imi_m — amplitude of current oscillations, A (ampere)
  • ω\omega — angular frequency, rad/s
  • qmq_m — amplitude of charge oscillations, C (coulomb)
Electromagnetic oscillations

Thomson oscillation period

T=2πLCT = 2\pi\sqrt{LC}
Symbols
  • TT — period of electromagnetic oscillations (Thomson period), s (second)
  • LL — inductance of the coil, H (henry)
  • CC — capacitance of the capacitor, F (farad)
Electromagnetic oscillations

Energy of the coil magnetic field

Wm=LI22W_m = \frac{LI^2}{2}
Symbols
  • WmW_m — energy of the magnetic field of the coil, J (joule)
  • LL — inductance of the coil, H (henry)
  • II — current in the coil, A (ampere)
Electromagnetic oscillations

Total oscillation energy

W=CU22+LI22W = \frac{CU^2}{2} + \frac{LI^2}{2}
Symbols
  • WW — total energy of the oscillatory circuit, J (joule)
  • CC — capacitance of the capacitor, F (farad)
  • UU — voltage across the capacitor, V (volt)
  • LL — inductance of the coil, H (henry)
  • II — current in the coil, A (ampere)
Electromagnetic oscillations

RMS current

Irms=Im2I_{\text{rms}} = \frac{I_m}{\sqrt{2}}
Symbols
  • IrmsI_{\text{rms}} — root-mean-square (effective) current, A (ampere)
  • ImI_m — peak (amplitude) alternating current, A (ampere)
Electromagnetic oscillations

RMS voltage

Urms=Um2U_{\text{rms}} = \frac{U_m}{\sqrt{2}}
Symbols
  • UrmsU_{\text{rms}} — root-mean-square (effective) voltage, V (volt)
  • UmU_m — peak (amplitude) alternating voltage, V (volt)
Electromagnetic oscillations

Inductive reactance

XL=ωL=2πfLX_L = \omega L = 2\pi f L
Symbols
  • XLX_L — inductive reactance, Ω (ohm)
  • ω\omega — angular frequency, rad/s
  • LL — inductance, H (henry)
  • ff — frequency of the alternating current, Hz (hertz)
Electromagnetic oscillations

Capacitive reactance

XC=1ωC=12πfCX_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}
Symbols
  • XCX_C — capacitive reactance, Ω (ohm)
  • ω\omega — angular frequency, rad/s
  • CC — capacitance, F (farad)
  • ff — frequency of the alternating current, Hz (hertz)
Electromagnetic oscillations

Total AC impedance

Z=R2+(XL−XC)2Z = \sqrt{R^2 + (X_L - X_C)^2}
Symbols
  • ZZ — total impedance of the AC circuit, Ω (ohm)
  • RR — resistance (ohmic), Ω (ohm)
  • XLX_L — inductive reactance, Ω (ohm)
  • XCX_C — capacitive reactance, Ω (ohm)
Electromagnetic oscillations

Ohm's law for an AC circuit

I=UZI = \frac{U}{Z}
Symbols
  • II — effective (RMS) current, A (ampere)
  • UU — effective (RMS) voltage, V (volt)
  • ZZ — total impedance of the circuit, Ω (ohm)