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A scalar field assigns a number to each point (temperature). A vector field assigns a vector (electric field, fluid velocity). Field lines are drawn tangent to the field, and their density shows its strength. Flux: the flux of a vector field through a surface is the surface integral of the field component normal to the surface, and measures the amount of field crossing it. Divergence: the flux per unit volume leaving an infinitesimal point. Positive divergence marks a source, negative a sink, and zero means no source. The divergence theorem links the flux out of a closed surface to the integral of the divergence inside. Medical relevance: electromagnetic waves are used in MRI, radiofrequency ablation, hyperthermia and tracking systems.

Charge comes in integer multiples of the elementary charge e = 1.6 x 10^-19 C and is conserved. Like charges repel and opposite charges attract. Coulomb's law: F = k q1 q2 / r^2, with k = 1/(4 pi epsilon0) about 9 x 10^9 N m^2/C^2. The force is along the line joining the charges. For several charges, forces add as vectors (superposition). Continuous distributions: charge is described by a density: volume density rho (C/m^3), surface density sigma (C/m^2) and linear density lambda (C/m). Total charge is the integral of the density.

Electric field: E = F/q. For a point charge E = k q / r^2 directed away from a positive charge. Gauss's law (integral form): the flux of E through a closed surface equals Q_enclosed / epsilon0. In differential form: divergence of E = rho / epsilon0. Use: for symmetric charge distributions Gauss's law gives E directly (sphere, infinite line, infinite plane). Inside a conductor in electrostatic equilibrium E = 0 and excess charge sits on the surface. Electric potential V: work per unit charge, E = minus gradient of V. Field lines are perpendicular to equipotential surfaces.

1. What does a positive divergence mean? 2. State Coulomb's law. 3. Which charge density describes a charged wire? 4. State Gauss's law in integral and differential forms. 5. What is the field inside a conductor in equilibrium?

1. The point is a source of the field. 2. F = q1 q2 / (4 pi epsilon0 r^2), directed along the line joining the charges. 3. Linear charge density (C/m). 4. Flux of E through a closed surface = Q_enclosed / epsilon0; divergence of E = rho / epsilon0. 5. Zero.

Current I is charge crossing a section per unit time (amperes). Current density J = n q v (A/m^2), a vector field; I is the flux of J through the section. Ohm's law: microscopic J = sigma E with conductivity sigma; macroscopic V = R I with R = L/(sigma A). Power dissipated: P = V I = R I^2 (Joule heating), which is the basis of resistive heating and electrosurgery. Conservation of charge: the divergence of J plus the rate of change of charge density is zero (the continuity equation). Semiconductors are doped to add carriers (n or p type).

Magnets have two poles that cannot be isolated. Oersted showed that a current deflects a compass, which links electricity and magnetism. Magnetic induction B (tesla) is defined by the force on a moving charge: F = q v x B (Lorentz force). Magnetic field lines are closed loops, so divergence of B = 0 (no magnetic monopoles). Ampere's law: the circulation of B around a closed path equals mu0 times the enclosed current. A long straight wire gives B = mu0 I / (2 pi r); the field inside a long solenoid is mu0 n I. Materials: paramagnetic, diamagnetic and ferromagnetic. Clinical tip: MRI magnets are 1.5-3 T, tens of thousands of times Earth's field.

1. Write the microscopic form of Ohm's law. 2. What is the continuity equation? 3. Why are magnetic field lines closed? 4. Give the field of a long solenoid.

1. J = sigma E. 2. Divergence of J plus d(rho)/dt equals zero: charge is conserved. 3. There are no magnetic monopoles, so divergence of B is zero. 4. B = mu0 n I.

Faraday's law: the emf around a closed loop equals minus the rate of change of magnetic flux through it: emf = -d(Phi)/dt. Differential form: curl E = -dB/dt. Lenz's law: the induced current creates a field that opposes the change in flux (the minus sign; conservation of energy). Applications: generators, transformers, inductors, induction heating, eddy currents. Medical: MRI gradients induce currents in tissue (peripheral nerve stimulation limits), and pickup coils detect the MRI signal.

1. Gauss (electric): divergence of E = rho/epsilon0. 2. Gauss (magnetic): divergence of B = 0. 3. Faraday: curl E = -dB/dt. 4. Ampere-Maxwell: curl B = mu0 J + mu0 epsilon0 dE/dt. Maxwell added the displacement current term (dE/dt) so that a changing electric field also creates a magnetic field. This makes possible self-sustaining electromagnetic waves travelling at c = 1/sqrt(mu0 epsilon0), about 3 x 10^8 m/s.

1. State Faraday's law. 2. What does Lenz's law say? 3. Which term did Maxwell add to Ampere's law and why?

1. emf = -d(flux)/dt. 2. The induced current opposes the change of flux that causes it. 3. The displacement current, so that changing electric fields produce magnetic fields and to keep charge conserved; it predicts electromagnetic waves.

A transmission line is modelled by series inductance and resistance and shunt capacitance and conductance per unit length. The telegraphist equations link voltage and current along the line and give wave solutions: forward and backward travelling waves in phasor form. Lossless line: characteristic impedance Z0 = sqrt(L/C) and propagation velocity 1/sqrt(LC). The reflection coefficient at a load ZL is (ZL - Z0)/(ZL + Z0); with a matched load there is no reflection. A mismatch creates standing waves, characterised by the standing wave ratio. Medical relevance: coaxial cables and microwave circuits connecting antennas and applicators.

In free space a plane wave has E and B perpendicular to each other and to the direction of propagation, in phase, with |E| = c |B|. Frequency f, wavelength lambda and speed satisfy c = f lambda. In a medium the speed is v = 1/sqrt(mu epsilon) and the wave is attenuated in lossy materials such as tissue (finite conductivity). Penetration (skin) depth decreases with frequency. At a boundary between two media part of the wave is reflected and part transmitted (Fresnel coefficients); at normal incidence the problem is equivalent to a transmission line problem, with wave impedances in place of characteristic impedances.

1. What are the two travelling waves on a transmission line? 2. When is there no reflection at the load? 3. How does penetration depth in tissue vary with frequency?

1. A forward wave (towards the load) and a backward (reflected) wave. 2. When the load impedance equals the characteristic impedance. 3. It decreases as frequency increases.

The simplest antenna is the Hertzian (short) dipole: a short current element that produces near fields that decay quickly and a far field that decays as 1/r and carries power away. Parameters: radiation pattern (angular distribution of power), directivity (peak intensity relative to an isotropic radiator), gain (directivity times efficiency), bandwidth and polarisation. Two antennas in free space are related by the Friis transmission formula. Biomedical uses: antennas for hyperthermia, wireless implants, body-area networks and tracking systems.

Radar (radio detection and ranging) sends a pulse and detects the echo. Range = c t / 2, where t is the round-trip delay. Angular resolution depends on the antenna beamwidth. The radar equation links received power to transmitted power, antenna gain, wavelength, and radar cross-section of the target, and falls with the fourth power of distance. History: the Chain Home system and the cavity magnetron. Medical applications: contactless monitoring of breathing and heartbeat and movement detection.

Optical sources: LEDs and lasers (coherent, narrow spectrum, directional; light amplification by stimulated emission). Dielectric guide: light in a high-index core surrounded by a lower-index cladding is confined by total internal reflection when the angle exceeds the critical angle sin(theta_c) = n2/n1. Modes, dispersion and attenuation limit bandwidth and length. Advantages of fibre: large bandwidth, low loss, immunity to electromagnetic interference, small size. Medical uses: endoscopy, laser surgery and photodynamic therapy.

1. Define directivity and gain of an antenna. 2. How is range obtained in radar? 3. What condition gives total internal reflection in a fibre?

1. Directivity: peak radiation intensity relative to an isotropic source. Gain: directivity multiplied by efficiency. 2. From the round-trip time delay: range = c t / 2. 3. The angle of incidence at the core-cladding interface must exceed the critical angle, requiring core index greater than cladding index.

Nuclei with spin (mainly hydrogen) have a magnetic moment. In a static field B0 they precess at the Larmor frequency f = gamma B0 / (2 pi); for hydrogen gamma/(2 pi) is 42.58 MHz/T. A radiofrequency pulse at the Larmor frequency tilts the magnetisation into the transverse plane; the decaying signal is the free induction decay. Relaxation: T1 (longitudinal recovery, spin-lattice) and T2 (transverse decay, spin-spin), different in each tissue, so weighting the sequence gives contrast between fat, water and lesions. The spin echo removes the effect of field inhomogeneity. Field homogeneity of B0 is essential. MRI adds gradient fields to encode position.

Electromagnetic description of tissue: conductivity and permittivity depend on tissue type (high for muscle, low for fat and bone) and frequency, so tissue absorbs and attenuates waves. SAR (specific absorption rate): power absorbed per unit mass (W/kg), SAR = sigma E^2 / (2 rho) for peak field. Safety limits are set for whole-body and local exposure to avoid excessive heating. Hyperthermia raises tumour temperature to about 41-45 degrees Celsius. Applicators: capacitive RF, inductive RF, and microwave radiative applicators; phased arrays focus the energy on the target by controlled interference. Radiofrequency ablation uses higher temperatures to destroy tissue locally.

Laser classes: class 1 and 2 are safe or safe by the blink reflex; class 3 can damage the eye; class 4 can cause skin and eye damage and fire. Medical uses: CO2 laser for skin resurfacing and cutting, other lasers for ablation and coagulation of targeted tissue. Electromagnetic compatibility (EMC) is the ability of a device to function in its environment without disturbing others. In hospitals the main dangers are radiated and conducted interference, electrostatic discharge and surges. Devices must pass EMC tests, but passing does not guarantee safety in every place of use. Simple mitigation includes distance from sources, shielding and proper cabling.

1. Give the Larmor frequency formula. 2. What are T1 and T2? 3. Define SAR. 4. What is the purpose of a phased array in hyperthermia? 5. What does EMC mean?

1. f = gamma B0 / (2 pi). 2. T1: longitudinal (spin-lattice) recovery time. T2: transverse (spin-spin) decay time. 3. Specific absorption rate: electromagnetic power absorbed per unit mass of tissue (W/kg). 4. To focus energy on the tumour by interference while sparing surrounding tissue. 5. Electromagnetic compatibility: functioning correctly in an electromagnetic environment without causing interference.