EM field theory and Maxwell equations

What to remember

• Maxwell's four equations unify electricity and magnetism: Gauss's law, Gauss's law for magnetism, Faraday's law and the Ampere-Maxwell law (which adds displacement current).
• In a uniform plane wave, E and H are perpendicular to each other and to the direction of travel; their ratio is the intrinsic impedance, 377 Ω in free space.
• Poynting vector S = E × H gives power flow per unit area; speed of light c = 1/√(μ0 ε0).

Static electric fields

Coulomb's law: F = Q1Q2/(4πε0 r²). Constant 1/(4πε0) ≈ 9 × 10⁹ N·m²/C². ε0 ≈ 8.854 × 10⁻¹² F/m. Electric field E = F/Q (V/m). Electric flux density D = εE (C/m²). Gauss's law: ∮ D·dS = Q enclosed; in point form ∇·D = ρv.

Potential: E = −∇V. Potential of a point charge V = Q/(4πε0 r). Work done in moving a charge: W = Q × (potential difference). Static E field is conservative: ∇ × E = 0.

Standard results: field of an infinite line charge ρL: E = ρL/(2πε0 r). Infinite sheet charge ρs: E = ρs/(2ε0), independent of distance. Field inside a conductor in electrostatics is zero. Capacitance: parallel plate C = εA/d; spherical (inner a, outer b): C = 4πε ab/(b − a); coaxial: C = 2πεL/ln(b/a). Energy stored: W = ½CV² = ½ε∫E² dv, with energy density ½εE².

Boundary conditions at the dielectric interface: the tangential component of E is continuous; the normal component of D is continuous (with no surface charge). At a conductor surface, E is normal and D = ρs. Poisson's equation: ∇²V = −ρv/ε. Laplace's equation: ∇²V = 0 (no charge).

Static magnetic fields

Biot-Savart law gives the field of a current element. Ampere's circuital law: ∮ H·dl = I enclosed. Standard results: infinite straight wire H = I/(2πr); centre of a circular loop of radius a: H = I/(2a); solenoid (long): H = nI inside, nearly zero outside. B = μH (Wb/m² = tesla), μ0 = 4π × 10⁻⁷ H/m. ∇·B = 0, so magnetic monopoles do not exist.

Force on a moving charge (Lorentz): F = Q(E + v × B). Force on a current element: F = I L × B. Force per unit length between two parallel wires carrying currents I1 and I2 at distance d: μ0 I1 I2/(2πd), attractive when the currents flow in the same direction. Boundary conditions for magnetic fields: normal B is continuous; tangential H is continuous (no surface current). Magnetic energy: W = ½LI²; energy density ½μH².

Time-varying fields: Maxwell's equations

Displacement current density Jd = ∂D/∂t. Maxwell added this term; it makes the current in a capacitor continuous and predicts electromagnetic waves. Conduction current J = σE. The ratio of conduction to displacement current in a medium is σ/(ωε); a good conductor has σ/(ωε) ≫ 1, a good dielectric has σ/(ωε) ≪ 1. Continuity equation: ∇·J = −∂ρv/∂t.

Faraday's law in integral form: emf = −dΦ/dt. Lenz's law fixes the sign: induced emf opposes the change causing it. Transformer emf comes from a time-varying B; motional emf comes from movement: e = (v × B)·l.

Uniform plane waves

In a source-free, lossless medium the wave equation is ∇²E = με ∂²E/∂t². Velocity v = 1/√(με) = c/√(μr εr). In free space c ≈ 3 × 10⁸ m/s. Wave number β = ω√(με) = 2π/λ. Wavelength λ = v/f.

Intrinsic impedance η = √(μ/ε). In free space η0 = √(μ0/ε0) ≈ 120π ≈ 377 Ω. In a dielectric, η = 377/√εr (for μr = 1). For example, εr = 4 gives η ≈ 188.5 Ω and v = c/2.

Properties: the wave is transverse electromagnetic (TEM); E and H are in phase in a lossless medium; E/H = η.

Polarisation: linear (E in a fixed direction), circular (two equal-amplitude components, 90° apart in phase), elliptical (the general case). A transmitting antenna vertical gives vertical polarisation.

Lossy media: propagation constant γ = α + jβ = √(jωμ(σ + jωε)). Attenuation constant α in Np/m; phase constant β in rad/m. Skin depth δ = 1/α; for a good conductor, δ = 1/√(π f μ σ). Skin depth decreases as frequency rises, which is why high-frequency currents flow on the conductor surface. The field falls to 1/e (37 %) of its surface value at one skin depth. In a good conductor, E leads H by 45° and η = (1 + j)/(σδ).

Reflection, power and radiation

A wave normally incident on a boundary between media 1 and 2 has reflection coefficient Γ = (η2 − η1)/(η2 + η1) and transmission coefficient τ = 2η2/(η2 + η1) = 1 + Γ. For a perfect conductor, η2 = 0, so Γ = −1 (total reflection) and a standing wave forms. VSWR = (1 + |Γ|)/(1 − |Γ|).

Poynting theorem: power flow density S = E × H (W/m²). Average power density for a plane wave: Sav = E²/(2η) for peak amplitude E, or Erms²/η. In free space, Erms²/377.

Example: if peak E = 37.7 V/m in free space, H = 37.7/377 = 0.1 A/m, Sav = ½ × 37.7 × 0.1 = 1.885 W/m².

Oblique incidence: Snell's law n1 sinθ1 = n2 sinθ2. Total internal reflection occurs beyond the critical angle, when going from a denser to a rarer medium. The Brewster angle is the angle at which a parallel-polarised wave has zero reflection: tanθB = √(ε2/ε1).

Radiation basics: an accelerating charge radiates. For a short dipole the far field varies as 1/r, and power density as 1/r². Radiation resistance of a short (Hertzian) dipole is proportional to (L/λ)². Free-space far-field region: E and H are in phase, ratio 377 Ω.

Vector tools and coordinate systems

The gradient of a scalar gives a vector pointing along the steepest increase. The divergence of a vector measures the net outward flux per unit volume (a source). The curl measures the circulation per unit area (a rotation). Divergence theorem: ∮ A·dS = ∫ (∇·A) dv. Stokes' theorem: ∮ A·dl = ∫ (∇×A)·dS. Three common coordinate systems are Cartesian (x, y, z), cylindrical (ρ, φ, z) and spherical (r, θ, φ); a line charge suits cylindrical coordinates and a point charge suits spherical coordinates. The curl of a gradient is always zero, and the divergence of a curl is always zero.

Potentials, inductance and magnetic circuits

The magnetic vector potential A is defined by B = ∇×A. Inductance L = Φ/I for one turn, or L = NΦ/I for N turns. Solenoid inductance: L = μN²A/l. Coaxial cable inductance per unit length: (μ/2π) ln(b/a). Mutual inductance M links two coils, and the coupling coefficient k = M/√(L1L2) is at most 1. The magnetic circuit analogy: mmf = NI (like emf), reluctance S = l/(μA) (like resistance), flux Φ (like current). Ferromagnetic materials show hysteresis, and the area of the B-H loop is the energy loss per cycle per unit volume. Permeability of ferromagnets is very high; of paramagnetic and diamagnetic materials it is close to μ0.

Worked examples

• 1. A wave in a medium with εr = 9, μr = 1: v = 3 × 10⁸/3 = 10⁸ m/s; η = 377/3 ≈ 125.7 Ω.
• 2. Wave of frequency 300 MHz in free space: λ = 3 × 10⁸/3 × 10⁸ = 1 m; β = 2π rad/m.
• 3. Normal incidence from air to a medium with η2 = 125.7 Ω: Γ = (125.7 − 377)/(125.7 + 377) = −0.5.
• 4. Line charge 10 nC/m at 2 m: E = 10⁻⁸/(2π × 8.854 × 10⁻¹² × 2) ≈ 90 V/m.

Exam traps

• The ratio E/H is η, not velocity; 377 Ω applies only to free space.
• Gauss's law for magnetism says ∇·B = 0 and not ∇·H = 0 for all materials.
• Displacement current is not a flow of charge; it is ∂D/∂t.
• Skin depth reduces with increasing frequency.
• Tangential E is continuous; normal D is continuous (no free surface charge).
• In a conductor the field inside is zero only for static fields.
• A perfect conductor gives Γ = −1, not +1.
• The sheet-charge field does not fall with distance.

One-liners

• 1. Speed of light c = 1/√(μ0 ε0).
• 2. η0 ≈ 377 Ω.
• 3. Faraday's law: ∇×E = −∂B/∂t.
• 4. Maxwell added the displacement current.
• 5. Velocity in a dielectric is c/√(μr εr).
• 6. Skin depth δ = 1/√(π f μ σ).
• 7. E and H of a plane wave are mutually perpendicular.
• 8. Poynting vector is E × H.
• 9. Laplace equation holds where charge density is zero.
• 10. Normal-incidence Γ = (η2 − η1)/(η2 + η1).
• 11. Brewster angle gives zero reflection for parallel polarisation.
• 12. B = μH.

Practice questions

1. Gauss's law in point form for electric flux density is

   1. ∇·D = ρv
   2. ∇·D = 0 always
   3. ∇·B = ρv
   4. ∇×D = ρv
   Answer

   A. ∇·D = ρv

   Divergence of D equals volume charge density.

2. Which Maxwell equation shows that magnetic monopoles do not exist?

   1. ∇·B = 0
   2. ∇·D = ρv
   3. ∇×E = −∂B/∂t
   4. ∇×H = J + ∂D/∂t
   Answer

   A. ∇·B = 0

   Net magnetic flux through a closed surface is zero.

3. Faraday's law in point form is

   1. ∇·E = ρ/ε
   2. ∇×B = 0
   3. ∇×E = −∂B/∂t
   4. ∇×H = J
   Answer

   C. ∇×E = −∂B/∂t

   A changing magnetic flux density induces an electric field.

4. The displacement current term in Maxwell's equations is

   1. ∂B/∂t
   2. ∂D/∂t
   3. ρv
   4. σE
   Answer

   B. ∂D/∂t

   Jd = ∂D/∂t.

5. Maxwell's modification of Ampere's law was necessary mainly to

   1. explain magnetic monopoles
   2. describe static charges only
   3. define the tesla
   4. make current continuity hold for capacitors and predict EM waves
   Answer

   D. make current continuity hold for capacitors and predict EM waves

   Without ∂D/∂t, ∇×H = J contradicts the continuity equation for time-varying fields.

6. The electric field of an infinite sheet of charge density ρs in free space is

   1. ρs/(2πε0 r)
   2. ρs/(2ε0), independent of distance
   3. ρs/ε0 times distance
   4. ρs/(4πε0 r²)
   Answer

   B. ρs/(2ε0), independent of distance

   Infinite sheet gives a uniform field.

7. The electric field of an infinite line charge ρL at distance r is

   1. ρL/(2πε0 r)
   2. ρL/(4πε0 r²)
   3. ρL r/(2πε0)
   4. ρL/(2ε0)
   Answer

   A. ρL/(2πε0 r)

   Gauss's law with a cylindrical surface.

8. A line charge of 2 nC/m lies in free space. Taking 1/(2πε0) = 18 × 10⁹, the field at 3 m is about

   1. 36 V/m
   2. 108 V/m
   3. 12 V/m
   4. 6 V/m
   Answer

   C. 12 V/m

   E = 18 × 10⁹ × 2 × 10⁻⁹/3 = 12 V/m.

9. The capacitance of a parallel plate capacitor is

   1. Aε d
   2. εd/A
   3. A/(εd)
   4. εA/d
   Answer

   D. εA/d

   Standard formula.

10. A parallel plate capacitor of area 0.01 m² and separation 1 mm in air has capacitance about (ε0 = 8.85 × 10⁻¹² F/m)

    1. 88.5 pF
    2. 8.85 pF
    3. 885 pF
    4. 0.885 pF
    Answer

    A. 88.5 pF

    C = 8.85 × 10⁻¹² × 0.01/10⁻³ = 8.85 × 10⁻¹¹ F.

11. The energy density of an electric field is

    1. ε/E
    2. ½ εE²
    3. εE²
    4. ½ E²/ε
    Answer

    B. ½ εE²

    Energy per unit volume.

12. At a boundary between two dielectrics, with no free surface charge, which component is continuous?

    1. Tangential component of B
    2. Tangential component of D
    3. Normal component of D
    4. Normal component of E
    Answer

    C. Normal component of D

    Tangential E and normal D are continuous.

13. Laplace's equation applies in a region where

    1. magnetic field is zero
    2. the medium is a conductor carrying current
    3. current density is infinite
    4. volume charge density is zero
    Answer

    D. volume charge density is zero

    Poisson's equation reduces to Laplace's when ρv = 0.

14. The magnetic field at distance r from a long straight wire carrying current I is

    1. I/(2πr)
    2. I r/(2π)
    3. μ0 I/r
    4. I/(4πr²)
    Answer

    A. I/(2πr)

    Ampere's circuital law.

15. A long straight wire carries 10 A. The magnetic field strength H at 0.5 m is nearly

    1. 31.8 A/m
    2. 3.18 A/m
    3. 20 A/m
    4. 6.37 A/m
    Answer

    B. 3.18 A/m

    H = 10/(2π × 0.5) = 3.18 A/m.

16. Two long parallel wires carry currents in the same direction. The force between them is

    1. repulsive
    2. zero
    3. perpendicular to the wires only
    4. attractive
    Answer

    D. attractive

    Parallel currents attract.

17. The Lorentz force on a charge Q moving with velocity v in fields E and B is

    1. Q(E − v·B)
    2. QvB only
    3. Q E × B
    4. Q(E + v × B)
    Answer

    D. Q(E + v × B)

    Standard electromagnetic force law.

18. The speed of an electromagnetic wave in free space equals

    1. √(μ0 ε0)
    2. μ0 ε0
    3. 1/√(μ0 ε0)
    4. √(μ0/ε0)
    Answer

    C. 1/√(μ0 ε0)

    c = 1/√(μ0 ε0).

19. The intrinsic impedance of free space is approximately

    1. 75 Ω
    2. 120 Ω
    3. 377 Ω
    4. 50 Ω
    Answer

    C. 377 Ω

    η0 = √(μ0/ε0) ≈ 120π Ω.

20. A uniform plane wave travels in a lossless medium with εr = 4 and μr = 1. Its velocity is

    1. 1.5 × 10⁸ m/s
    2. 0.75 × 10⁸ m/s
    3. 6 × 10⁸ m/s
    4. 3 × 10⁸ m/s
    Answer

    A. 1.5 × 10⁸ m/s

    v = c/√4 = c/2.

21. The intrinsic impedance of a lossless dielectric with εr = 4 and μr = 1 is about

    1. 754 Ω
    2. 188.5 Ω
    3. 377 Ω
    4. 94 Ω
    Answer

    B. 188.5 Ω

    η = 377/2 = 188.5 Ω.

22. The wavelength of a 150 MHz wave in free space is

    1. 1 m
    2. 0.5 m
    3. 2 m
    4. 20 m
    Answer

    C. 2 m

    λ = 3 × 10⁸/1.5 × 10⁸ = 2 m.

23. In a uniform plane wave in a lossless medium, the electric and magnetic fields are

    1. parallel and in phase
    2. perpendicular and 90° out of phase
    3. parallel and 90° out of phase
    4. perpendicular to each other and in phase
    Answer

    D. perpendicular to each other and in phase

    TEM wave with E/H = η.

24. Skin depth in a good conductor

    1. decreases as frequency increases
    2. is infinite for a perfect conductor
    3. is independent of conductivity
    4. increases as frequency increases
    Answer

    A. decreases as frequency increases

    δ = 1/√(π f μ σ).

25. At a distance of one skin depth, the field amplitude falls to

    1. 1/10 of its surface value
    2. 1/e of its surface value
    3. 1/2 of its surface value
    4. zero
    Answer

    B. 1/e of its surface value

    E = E0 e^(−z/δ).

26. The unit of the attenuation constant α is

    1. siemens
    2. neper per metre
    3. ohm
    4. radian per metre
    Answer

    B. neper per metre

    α is in Np/m; β in rad/m.

27. The reflection coefficient at a perfect conductor surface for normal incidence is

    1. −1
    2. 0
    3. +1
    4. +0.5
    Answer

    A. −1

    η2 = 0 gives Γ = −1.

28. A plane wave in air meets a medium with intrinsic impedance 125.7 Ω. The normal-incidence reflection coefficient is about

    1. −0.25
    2. −1
    3. +0.5
    4. −0.5
    Answer

    D. −0.5

    (125.7 − 377)/(125.7 + 377) ≈ −0.5.

29. The VSWR for a reflection coefficient of magnitude 0.5 is

    1. 2
    2. 1.5
    3. 0.5
    4. 3
    Answer

    D. 3

    (1 + 0.5)/(1 − 0.5) = 3.

30. Poynting vector represents

    1. force per unit length
    2. energy stored per unit volume
    3. power flow per unit area
    4. charge density
    Answer

    C. power flow per unit area

    S = E × H in W/m².

31. A plane wave in free space has rms electric field 37.7 V/m. The average power density is

    1. 377 W/m²
    2. 3.77 W/m²
    3. 0.377 W/m²
    4. 37.7 W/m²
    Answer

    B. 3.77 W/m²

    E²/η = 1421/377 ≈ 3.77.

32. The Brewster angle is the angle of incidence at which

    1. reflection of parallel-polarised wave is zero
    2. transmission is zero
    3. total internal reflection begins
    4. the wave is circularly polarised
    Answer

    A. reflection of parallel-polarised wave is zero

    At Brewster angle, parallel polarisation is fully transmitted.

33. Total internal reflection can occur when a wave travels from

    1. air to glass at normal incidence
    2. a rarer to a denser medium
    3. a denser medium to a rarer medium at an angle beyond the critical angle
    4. any medium at 0° incidence
    Answer

    C. a denser medium to a rarer medium at an angle beyond the critical angle

    Needs n1 > n2.

34. Consider the statements. 1. In free space, E and H of a plane wave are in phase. 2. Intrinsic impedance of free space is about 377 Ω. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both are correct.

35. Consider the statements. 1. A static electric field is conservative. 2. The curl of a static E field is non-zero. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    For static fields ∇×E = 0.

36. Consider the statements. 1. Displacement current exists only in conductors. 2. Displacement current density is ∂D/∂t. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    It exists even in vacuum or dielectrics.

37. Consider the statements. 1. A good conductor has σ/(ωε) much greater than 1. 2. Skin depth increases as frequency increases. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Skin depth decreases with frequency.

38. Consider the statements. 1. Tangential component of E is continuous across a boundary. 2. Normal component of B is continuous across a boundary. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both are standard boundary conditions.

39. Consider the statements about the Poynting vector. 1. It equals E × H. 2. It points in the direction of wave propagation. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both are true for a plane wave.

40. Consider the statements. 1. Inside a conductor under electrostatic conditions, E = 0. 2. The field of an infinite sheet charge falls as 1/r. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Sheet-charge field is uniform.

41. Match the Maxwell equation with its name: P. ∇·D = ρv Q. ∇×E = −∂B/∂t R. ∇×H = J + ∂D/∂t 1. Faraday 2. Ampere-Maxwell 3. Gauss

    1. P-2, Q-1, R-3
    2. P-1, Q-3, R-2
    3. P-3, Q-2, R-1
    4. P-3, Q-1, R-2
    Answer

    D. P-3, Q-1, R-2

    Standard names.

42. Match the quantity with its unit: P. D Q. B R. H 1. A/m 2. C/m² 3. tesla

    1. P-1, Q-3, R-2
    2. P-2, Q-3, R-1
    3. P-3, Q-2, R-1
    4. P-2, Q-1, R-3
    Answer

    B. P-2, Q-3, R-1

    D in C/m², B in tesla, H in A/m.

43. Match the formula with the result: P. Free-space speed Q. Intrinsic impedance R. Skin depth 1. √(μ/ε) 2. 1/√(π f μ σ) 3. 1/√(μ0 ε0)

    1. P-3, Q-2, R-1
    2. P-2, Q-1, R-3
    3. P-1, Q-3, R-2
    4. P-3, Q-1, R-2
    Answer

    D. P-3, Q-1, R-2

    Standard expressions.

44. Consider the statements about wave polarisation. 1. Circular polarisation needs two equal-amplitude components 90° apart in phase. 2. In linear polarisation, E keeps a fixed direction. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both definitions are standard.

45. Consider the statements. 1. Lenz's law states that the induced emf opposes the cause producing it. 2. Gauss's law for magnetism says ∇·B = ρ. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    ∇·B = 0, so statement 2 is false.