Electromagnetism Formulas

Force and Field Formulas

Coulomb's law is the starting point for electrostatics: F = kQ1Q2/r squared, where k is Coulomb's constant (approximately 8.99 times 10 to the 9th N m squared per C squared), Q1 and Q2 are the magnitudes of the two charges, and r is the distance between them. The force is attractive for opposite charges and repulsive for like charges. The electric field created by a point charge is E = kQ/r squared, representing the force per unit charge at distance r from the source charge.

The Lorentz force law gives the total electromagnetic force on a charged particle: F = qE + qv cross B, where q is the particle's charge, E is the electric field, v is the particle's velocity, and B is the magnetic field. The electric force acts in the direction of the field (or opposite for negative charges), while the magnetic force is perpendicular to both the velocity and the field. This formula is the foundation for understanding particle motion in electromagnetic fields, from cathode ray tubes to cyclotrons.

The force on a current-carrying wire in a magnetic field is F = IL cross B, where I is the current, L is the length vector of the wire, and B is the magnetic field. This formula governs the operation of electric motors. The torque on a current loop is tau = nIA cross B, where n is the number of turns and A is the area vector of the loop. These force and torque equations connect electromagnetic theory to mechanical engineering.

Circuit and Component Formulas

Ohm's law, V = IR, is the most frequently used equation in electrical engineering. It states that the voltage across a resistor equals the current through it multiplied by its resistance. Rearranged, it gives I = V/R for finding current and R = V/I for finding resistance. For circuits with multiple resistors, series resistances add (R_total = R1 + R2 + R3) and parallel resistances combine by reciprocals (1/R_total = 1/R1 + 1/R2 + 1/R3).

Capacitance is defined as C = Q/V, the ratio of stored charge to applied voltage. For a parallel-plate capacitor, C = epsilon A/d, where epsilon is the permittivity of the dielectric material, A is the plate area, and d is the plate separation. The energy stored in a capacitor is E = 1/2 CV squared. In AC circuits, the capacitive reactance is Xc = 1/(2 pi fC), which decreases with increasing frequency.

Inductance relates magnetic flux linkage to current: L = N phi / I, where N is the number of turns and phi is the magnetic flux through each turn. The energy stored in an inductor is E = 1/2 LI squared. In AC circuits, the inductive reactance is XL = 2 pi fL, which increases with increasing frequency. The resonant frequency of an LC circuit is f = 1/(2 pi sqrt(LC)), where inductive and capacitive reactances are equal.

Power and Energy Formulas

Electrical power is P = IV (power equals current times voltage). Combining with Ohm's law yields two additional forms: P = I squared R (useful when current and resistance are known) and P = V squared / R (useful when voltage and resistance are known). These three forms of the power equation cover virtually every practical power calculation in DC circuits.

In AC circuits, power calculations must account for the phase relationship between voltage and current. Real power (watts) is P = IV cos(phi), where phi is the phase angle. Reactive power (volt-amperes reactive, or VAR) is Q = IV sin(phi). Apparent power (volt-amperes, or VA) is S = IV. The power factor, cos(phi), indicates how effectively the circuit converts apparent power into real work. A power factor of 1.0 (purely resistive load) means all power does useful work; lower power factors indicate energy cycling between the source and reactive components.

Energy is power integrated over time: E = Pt for constant power. The kilowatt-hour (kWh), equal to 3.6 megajoules, is the standard unit for electrical energy billing. A 1000-watt appliance running for one hour consumes one kilowatt-hour. The energy stored in electric and magnetic fields is given by u_E = 1/2 epsilon E squared (energy density in an electric field) and u_B = 1/2 B squared / mu_0 (energy density in a magnetic field).

Electromagnetic Induction Formulas

Faraday's law of electromagnetic induction states that the induced electromotive force (EMF) in a loop equals the negative rate of change of magnetic flux through the loop: EMF = -d(phi_B)/dt. For a coil with N turns, EMF = -N d(phi_B)/dt. The negative sign (Lenz's law) indicates that the induced EMF opposes the change in flux that produced it, consistent with conservation of energy.

Magnetic flux is phi_B = B dot A = BA cos(theta), where B is the magnetic field, A is the area of the loop, and theta is the angle between the field direction and the normal to the loop. Flux changes (and therefore induced EMFs) can result from changing the field strength, changing the loop area, or changing the angle between them. Generators exploit the changing angle as a coil rotates in a magnetic field.

Transformer equations relate primary and secondary voltages and currents through the turns ratio: V_s/V_p = N_s/N_p and I_s/I_p = N_p/N_s. An ideal transformer conserves power (V_p I_p = V_s I_s), so stepping up voltage proportionally steps down current and vice versa. Mutual inductance M between two coils relates the EMF induced in one coil to the rate of current change in the other: EMF_2 = -M dI_1/dt.

Electromagnetic Wave Formulas

The speed of electromagnetic waves in vacuum is c = 1/sqrt(mu_0 epsilon_0), where mu_0 is the permeability of free space and epsilon_0 is the permittivity of free space. This gives c = 299,792,458 meters per second. The relationship between wavelength, frequency, and speed is c = f lambda. In a material with relative permittivity epsilon_r and relative permeability mu_r, the wave speed is reduced to v = c/sqrt(epsilon_r mu_r).

The energy carried by a photon is E = hf = hc/lambda, where h is Planck's constant (6.626 times 10 to the negative 34th joule-seconds). The intensity of an electromagnetic wave (power per unit area) is given by the Poynting vector S = (1/mu_0) E cross B, and for a sinusoidal wave the time-averaged intensity is S_avg = E_0 B_0 / (2 mu_0) = c epsilon_0 E_0 squared / 2.

The radiation pressure exerted by electromagnetic waves on a surface is p = S/c for complete absorption and p = 2S/c for perfect reflection, where S is the intensity. While this pressure is extremely small for ordinary light, it is significant in astrophysics (radiation pressure from stars drives stellar winds) and is exploited in proposed solar sail spacecraft that would use sunlight pressure for propulsion.