Electromagnetism
Electromagnetism is the branch of physics that describes the interactions of electric charges and currents through the electromagnetic field. The 19th-century synthesis of electricity, magnetism and optics by James Clerk Maxwell stands among the greatest intellectual achievements of physics; his four equations underpin every modern technology from radio to MRI.
The force per unit positive test charge at a point in space: E = F/q. It is a vector field whose direction at any point gives the direction of force on a positive charge placed there. SI unit: newton per coulomb (N/C) or volt per metre (V/m).
Electrostatics
The Coulomb's law (1785) states that the force between two point charges is
F = (1/4πε₀) · q₁q₂ / r²
where ε₀ = 8.854 × 10⁻¹² C²/(N·m²) is the permittivity of free space. The force is repulsive for like charges and attractive for unlike charges, and acts along the line joining them.
The electric potential V at a distance r from a point charge q is V = q/(4πε₀r), and the potential energy of two charges is U = q₁q₂/(4πε₀r).
Gauss's law
The flux of the electric field through any closed surface equals the enclosed charge divided by ε₀:
∮ E · dA = Q_enclosed / ε₀
This integral form is mathematically equivalent to Coulomb's law but is far more useful for symmetric charge distributions (spheres, cylinders, infinite planes).
Capacitance and dielectrics
A capacitor stores charge Q at a potential difference V; its capacitance is C = Q/V (farads). For a parallel-plate capacitor in vacuum, C = ε₀A/d. Inserting a dielectric of relative permittivity εᵣ multiplies the capacitance by εᵣ. The energy stored is U = ½CV² = ½Q²/C.
- Charge is quantised (e = 1.602 × 10⁻¹⁹ C) and conserved.
- Electric field lines start on positive charges and end on negative ones; they never cross.
- Inside an ideal conductor in electrostatic equilibrium, E = 0 and excess charge resides on the surface.
- A uniformly charged spherical shell behaves externally as a point charge at its centre.
Magnetism and steady currents
Moving charges produce magnetic fields. The Biot–Savart law gives the field dB produced by a current element I dl:
dB = (μ₀/4π) · I dl × r̂ / r²
where μ₀ = 4π × 10⁻⁷ T·m/A is the permeability of free space. For an infinitely long straight wire carrying current I, B = μ₀I/(2πr).
A charged particle moving with velocity v in a magnetic field B experiences the Lorentz force:
F = qv × B (magnetic) or F = q(E + v × B) (combined)
Because this force is always perpendicular to v, magnetic forces do no work; they only deflect charges, giving rise to circular or helical motion.
Ampère's law
The integral of the magnetic field around any closed loop equals μ₀ times the enclosed current: ∮ B · dl = μ₀ I_enc. Together with Gauss's law, it provides the simplest method to find fields of symmetric current distributions (solenoid, toroid, coaxial cable).
Electromagnetic induction
Michael Faraday (1831) discovered that a changing magnetic flux through a loop induces an EMF:
EMF = − dΦ_B / dt
The minus sign embodies Lenz's law: the induced current opposes the change that produced it (an expression of energy conservation). Faraday's discovery led directly to the electric generator, transformer and induction motor — the basis of the modern electric power industry.
Maxwell's equations
Maxwell unified the laws of electricity and magnetism by adding the displacement current term to Ampère's law. The four equations of classical electromagnetism in vacuum are:
| Equation | Differential form | Physical meaning |
|---|---|---|
| Gauss (E) | ∇·E = ρ/ε₀ | Charges create diverging E-field |
| Gauss (B) | ∇·B = 0 | No magnetic monopoles |
| Faraday | ∇×E = −∂B/∂t | Changing B induces E |
| Ampère–Maxwell | ∇×B = μ₀J + μ₀ε₀ ∂E/∂t | Currents and changing E create B |
Solving these in free space yields wave equations with propagation speed c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s — Maxwell thereby identified light as an electromagnetic wave.
For competitive exams, memorise the constants and the four Maxwell equations by name. A common question asks which equation expresses the non-existence of magnetic monopoles — the answer is Gauss's law for magnetism (∇·B = 0).
The electromagnetic spectrum
Electromagnetic waves span a vast range of frequencies, all travelling at c in vacuum:
| Region | Wavelength | Source / Use |
|---|---|---|
| Radio | > 1 m | Broadcasting, telecom |
| Microwave | 1 mm – 1 m | Radar, cooking, Wi-Fi |
| Infrared | 700 nm – 1 mm | Heat radiation, remote controls |
| Visible | 400 – 700 nm | Human vision |
| Ultraviolet | 10 – 400 nm | Sterilisation, sunburn |
| X-rays | 0.01 – 10 nm | Medical imaging |
| Gamma rays | < 0.01 nm | Nuclear processes, radiotherapy |
The same equations describe all of them — a triumph of theoretical unification.