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Unit 1: Electromagnetic Theory

PHY109 — Engineering Physics 9 min read

I. Orientation—Fields and Sources

A. Electromagnetic Framework

Electromagnetic theory describes how electric charges and currents produce electric and magnetic fields and how changing fields propagate through space as electromagnetic waves.

  • Governing idea: Charges are sources of electric fields, currents are sources of magnetic fields, and time-varying electric and magnetic fields generate each other.
  • Field viewpoint: A field assigns a physical quantity to every point ((x,y,z)) and time (t), replacing action-at-a-distance with local interactions.
  • SI conventions:
    • Electric field (\mathbf{E}): volt per metre ((\mathrm{V\,m^{-1}})).
    • Electric displacement (\mathbf{D}): coulomb per square metre ((\mathrm{C\,m^{-2}})).
    • Magnetic flux density (\mathbf{B}): tesla ((\mathrm{T})).
    • Magnetic field intensity (\mathbf{H}): ampere per metre ((\mathrm{A\,m^{-1}})).
    • Charge density (\rho): coulomb per cubic metre ((\mathrm{C\,m^{-3}})).
    • Current density (\mathbf{J}): ampere per square metre ((\mathrm{A\,m^{-2}})).
  • Material relations: In a linear, homogeneous, isotropic medium,
    TEXT
      D = εE,        B = μH,        J = σE

    where (\varepsilon) is permittivity, (\mu) is permeability, and (\sigma) is conductivity.
  • Mathematical language: The operator (\nabla) produces gradient, divergence, and curl; integral theorems connect local field behavior with behavior over finite regions.

II. Fields—Spatially Distributed Quantities

A. Scalar and vector fields

Scalar and vector fields distinguish quantities having magnitude alone from those having both magnitude and direction.

  • Scalar field: A function (\phi(x,y,z,t)) assigns one number to each space-time point; examples include electric potential (V), charge density (\rho), and temperature (T).
  • Vector field: A function assigns a vector to every point:
    TEXT
      A(x,y,z,t) = Ax i + Ay j + Az k

    where (A_x,A_y,A_z) are Cartesian components and (\mathbf{i},\mathbf{j},\mathbf{k}) are unit vectors.
  • Electromagnetic examples: (\mathbf{E}), (\mathbf{D}), (\mathbf{B}), (\mathbf{H}), and (\mathbf{J}) are vector fields because their direction affects force, flux, or current flow.
  • Field representation: Scalar fields are visualized using equal-value surfaces, such as equipotential surfaces; vector fields use arrows or field lines tangent to the local vector.
  • Superposition: For linear media, the total field equals the vector sum of fields produced by individual sources.

III. Differential Field Operators—Local Variation

A. Concept of gradient, divergence and curl

Gradient, divergence, and curl describe three distinct forms of spatial variation: maximum increase, net outward flow, and local circulation.

  1. Gradient:

    TEXT
       grad φ = ∇φ
          = i(∂φ/∂x) + j(∂φ/∂y) + k(∂φ/∂z)
    • Meaning: (\nabla\phi) points in the direction of the greatest increase of scalar field (\phi); its magnitude is the maximum rate of change per unit distance.
    • Electrostatic relation: Electric potential decreases along the electric field:
      TEXT
           E = −∇V

      Thus, (\mathbf{E}) is normal to equipotential surfaces and points toward decreasing (V).

  2. Divergence:

    TEXT
       div A = ∇·A
         = ∂Ax/∂x + ∂Ay/∂y + ∂Az/∂z
    • Meaning: (\nabla\cdot\mathbf{A}) measures the net outward flux per unit volume around a point.
    • Interpretation: Positive divergence indicates a source, negative divergence a sink, and zero divergence a locally source-free field.
    • Example: (\nabla\cdot\mathbf{B}=0) means magnetic flux has no isolated source or sink.

  3. Curl:

    TEXT
       curl A = ∇×A
          = i(∂Az/∂y − ∂Ay/∂z)
          + j(∂Ax/∂z − ∂Az/∂x)
          + k(∂Ay/∂x − ∂Ax/∂y)
    • Meaning: (\nabla\times\mathbf{A}) measures local circulation or rotational tendency; its direction follows the right-hand rule.
    • Irrotational field: If (\nabla\times\mathbf{A}=0) in a simply connected region, (\mathbf{A}) can be expressed as the gradient of a scalar potential.
    • Identity: The curl of any gradient and divergence of any curl vanish:
      TEXT
           ∇×(∇φ) = 0,        ∇·(∇×A) = 0

IV. Integral Theorems—Local and Global Descriptions

A. Gauss theorem and Stokes theorem (qualitative)

Gauss and Stokes theorems convert volume or surface information into equivalent boundary information, linking differential and integral field equations.

  1. Gauss divergence theorem:

    TEXT
       ∮S A·dS = ∫V (∇·A)dV
    • Statement: The total outward flux of (\mathbf{A}) through closed surface (S) equals the integral of its divergence over enclosed volume (V).
    • Elements: (d\mathbf{S}=\hat{\mathbf{n}}\,dS), where (\hat{\mathbf{n}}) is the outward unit normal; (dV) is a volume element.
    • Use: It converts (\nabla\cdot\mathbf{D}=\rho) into electric flux equal to enclosed charge.

  2. Stokes theorem:

    TEXT
       ∮C A·dl = ∫S (∇×A)·dS
    • Statement: Circulation of (\mathbf{A}) around closed contour (C) equals the flux of its curl through any surface (S) bounded by (C).
    • Orientation: The direction of line element (d\mathbf{l}) and surface normal (d\mathbf{S}) are related by the right-hand rule.
    • Use: It connects circulation laws, such as Faraday’s and Ampere’s laws, with their differential forms.

V. Potential Equations—Fields from Charge Distributions

A. Poisson and Laplace equations

Poisson and Laplace equations determine electrostatic potential from charge density and boundary conditions.

  • Derivation: Combining (\mathbf{E}=-\nabla V), (\mathbf{D}=\varepsilon\mathbf{E}), and (\nabla\cdot\mathbf{D}=\rho) gives, for constant (\varepsilon),
    TEXT
      ∇²V = −ρ/ε

    where (V) is electric potential and (\nabla^2) is the Laplacian operator.
  • Poisson equation: The equation (\nabla^2V=-\rho/\varepsilon) applies in a region containing volume charge.
  • Laplacian in Cartesian coordinates:
    TEXT
      ∇²V = ∂²V/∂x² + ∂²V/∂y² + ∂²V/∂z²
  • Laplace equation: In a charge-free region, (\rho=0), so
    TEXT
      ∇²V = 0
  • Boundary conditions: A unique solution is obtained by specifying potential on the boundary or specifying its normal derivative consistently.
  • Physical implication: A solution of Laplace’s equation cannot possess an isolated maximum or minimum inside a charge-free region; extrema occur at boundaries.

VI. Conservation of Charge

A. Continuity equation

The continuity equation expresses the local and global conservation of electric charge.

  • Integral form:
    TEXT
      ∮S J·dS = −d/dt ∫V ρ dV

    The outward current through closed surface (S) equals the rate at which charge decreases inside volume (V).
  • Differential form:
    TEXT
      ∇·J = −∂ρ/∂t

    It follows from the integral form by applying Gauss’s theorem.
  • Steady current: If (\partial\rho/\partial t=0), then (\nabla\cdot\mathbf{J}=0); no net charge accumulates at a point.
  • Significance: The negative sign means that positive outward current corresponds to a reduction of enclosed charge.

VII. Magnetostatic Circulation

A. Ampere Circuital Law

Ampere’s circuital law relates the circulation of a steady magnetic field to the conduction current enclosed by a closed path.

  • Integral form:
    TEXT
      ∮C H·dl = Ienclosed = ∫S J·dS

    where (I_{\text{enclosed}}) is the conduction current passing through surface (S).
  • Differential form: Applying Stokes theorem gives
    TEXT
      ∇×H = J
  • Right-hand rule: If curled fingers follow the integration path, the thumb gives the positive current direction through the surface.
  • Symmetric example: Around a long straight wire carrying current (I), circular symmetry gives
    TEXT
      H = I/(2πr)

    where (r) is radial distance from the wire.
  • Limitation: The original law is valid for steady currents but becomes inconsistent where charge or electric flux changes with time.

VIII. Time-Varying Electric Flux

A. Maxwell displacement current and correction in Ampere Circuital Law

Maxwell introduced displacement current so that Ampere’s law remains consistent with charge conservation in time-varying systems.

  • Capacitor problem: Conduction current flows in connecting wires, but no conduction current crosses the dielectric gap; the original law would give different results for different surfaces bounded by the same contour.
  • Displacement current density:
    TEXT
      Jd = ∂D/∂t

    where (\mathbf{J}_d) represents the magnetic effect of a changing electric displacement field.
  • Displacement current:
    TEXT
      Id = d/dt ∫S D·dS

    For an ideal charging capacitor, (I_d) in the gap equals the conduction current in the wires.
  • Corrected Ampere law:
    TEXT
      ∮C H·dl = ∫S (J + ∂D/∂t)·dS
  • Differential form:
    TEXT
      ∇×H = J + ∂D/∂t
  • Consistency: Taking divergence and using (\nabla\cdot(\nabla\times\mathbf{H})=0) yields (\nabla\cdot\mathbf{J}=-\partial\rho/\partial t), because (\nabla\cdot\mathbf{D}=\rho).

IX. Unified Electromagnetic Laws

A. Maxwell electromagnetic equations (differential and integral forms)

Maxwell’s four equations unify electric fields, magnetic fields, charges, currents, and electromagnetic induction.

  1. Gauss’s law for electricity:

    TEXT
       Differential:  ∇·D = ρ
   Integral:      ∮S D·dS = Qenclosed


    Electric charge (Q_{\text{enclosed}}=\int_V\rho\,dV) is the source of electric flux.

  2. Gauss’s law for magnetism:

    TEXT
       Differential:  ∇·B = 0
   Integral:      ∮S B·dS = 0


    Magnetic field lines form closed loops; no isolated magnetic monopoles occur in classical electromagnetism.

  3. Faraday’s law of electromagnetic induction:

    TEXT
       Differential:  ∇×E = −∂B/∂t
   Integral:      ∮C E·dl = −d/dt ∫S B·dS


    A changing magnetic flux creates a circulating electric field; the minus sign expresses Lenz’s law.

  4. Ampere–Maxwell law:

    TEXT
       Differential:  ∇×H = J + ∂D/∂t
   Integral:      ∮C H·dl = ∫S J·dS + d/dt ∫S D·dS


    Magnetic circulation is produced by conduction current and changing electric flux.

X. Interpretation of Maxwell’s Theory

A. Physical significance of Maxwell equations

The physical significance of Maxwell’s equations lies in their unified description of electromagnetic sources, induction, conservation, and wave propagation.

  • Source structure: Electric charges produce divergent electric fields, while the zero divergence of (\mathbf{B}) requires continuous magnetic field lines.
  • Mutual generation: Time-varying (\mathbf{B}) produces circulating (\mathbf{E}), and time-varying (\mathbf{D}) produces circulating (\mathbf{H}).
  • Electromagnetic waves: In a source-free homogeneous medium, the equations imply
    TEXT
      ∇²E = με ∂²E/∂t²,        ∇²H = με ∂²H/∂t²

    with wave speed (v=1/\sqrt{\mu\varepsilon}); in vacuum, (v=c=1/\sqrt{\mu_0\varepsilon_0}).
  • Field orientation: In a uniform plane wave, (\mathbf{E}), (\mathbf{H}), and the propagation direction are mutually perpendicular.
  • Technological scope: The equations govern antennas, radio transmission, optical propagation, transformers, waveguides, electric machines, and electromagnetic energy transfer.
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