Electric and magnetic field formulas

14 standard field results, each with a derivation and the points exams test.

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Field due to	Formula
Electric field due to a point charge	E = kq/r² = q/(4πε₀r²)
Electric field due to an electric dipole on its axis	E = 2kp/r³
Electric field due to an electric dipole on its equatorial line	E = kp/r³
Electric field due to a charged ring on its axis	E = kQx/(x² + R²)^(3/2)
Electric field due to an infinite line charge	E = λ/(2πε₀r) = 2kλ/r
Electric field due to an infinite charged sheet	E = σ/(2ε₀)
Electric field due to a charged conducting sphere (or spherical shell)	E = kQ/r² outside, 0 inside
Electric field due to a uniformly charged non-conducting sphere	E = kQr/R³ inside, kQ/r² outside
Magnetic field due to a long straight wire	B = μ₀I/(2πr)
Magnetic field due to a finite straight wire	B = (μ₀I/4πd)(sin α + sin β)
Magnetic field due to a circular loop at its centre	B = μ₀I/(2R)
Magnetic field due to a circular loop on its axis	B = μ₀IR²/(2(R² + x²)^(3/2))
Magnetic field due to a long solenoid	B = μ₀nI
Magnetic field due to a toroid	B = μ₀NI/(2πr)

Which law to use

Gauss's law (∮E · dA = q/ε₀) for symmetric charge: spheres, infinite lines and sheets.
Coulomb's law with superposition for point charges, dipoles and rings.
Ampère's law (∮B · dl = μ₀I) for symmetric currents: long wires, solenoids, toroids.
Biot–Savart law (dB = (μ₀/4π) I dl × r̂/r²) for loops, arcs and finite wires.

What is the electric field due to an infinite sheet?

σ/2ε₀, the same at every distance.

What is the magnetic field at the centre of a circular loop?

μ₀I/2R, or μ₀NI/2R for N turns.

What is the magnetic field inside a solenoid?

μ₀nI, where n is turns per unit length.

What is the electric field inside a conducting sphere?

Zero. All the charge sits on the surface.