Electromagnetic Waves and Fields

Potentials of Electromagnetic Waves and Fields

• Course code for this topic is TCE 304.
• Maxwell's Equations describe how electric and magnetic fields interact and propagate.
• Electromagnetic potentials (φ and A) offer an alternative way to express fields.
• Poynting vector quantifies energy flow in electromagnetic fields.

Electromagnetic Waves

• Electromagnetic waves are solutions to Maxwell's equations, and are a disturbance in the electric and magnetic field that propagates through space.
• Electromagnetic waves transfer energy from one point to another.
• The magnetic and electric fields of EM waves are perpendicular to each other and to the direction of the wave.
• EM waves are transverse in nature.
• The wave equation is derived from Maxwell's equations.
• EM waves propagate in a vacuum with a velocity C and transport energy and momentum through space.
• Permeability of magnetic field in free space is 4π.
• Permittivity of electric field in free space = 4π.
• The speed of EM waves is much lower in media different from vacuum due to transmission and absorption phenomena.
• Scalar potentials (φ, A) simplify electromagnetic wave representation.
  • E = -∇φ - ∂A/∂t
  • B = ∇×A
• Scalar and vector potentials are functions of position and time.
• The electric field in the presence of a static charge distribution p(r') is found from Coulomb's law:
  • E(r) = (1 / 4πε₀) ∫ [p(r') (r - r')] / |r - r'|³ dV'
• In terms of the scalar potential, for a static charge distribution the equation is:
  • φ(r) = (1 / 4πε₀) ∫ p(r') / |r - r'| dV'
• Calculating the potential is simpler than calculating the field directly.
• From Maxwell's equations it follows that in a homogeneous, isotropic medium the equation is:
  • ∇ ⋅ E = -∇ ⋅ ∇φ = p / ε
• Poisson's equation is:
  • ∇²φ(r) = -p(r) / ε
• In the presence of sources for the magnetic field, the magnetic field B can be found from the Biot-Savart law:
  • B(r) = (μ₀ / 4π) ∫ [J(r') × (r - r')] / |r - r'|³ dV'
• In a static case with constant fields, charges and currents, the magnetic field is related to the current density by:
  • ∇ × B = μJ
• After substituting B = ∇ × A and using the vector identity, the equation is:
  • ∇²A - ∇ (∇ ⋅ A) = -μJ

Electromagnetic Fields

• Field potentials allow alternative representations of E and B.
• Vector potential (A) and scalar potential (φ) simplify Maxwell’s equations.
• Field potentials can be applied to solving boundary-value problems
• Field potentials are used in quantum mechanics (Aharonov-Bohm effect).

Poynting Vector

• Poynting Theorem is about the conservation of energy density.
• The theorem states that: ∂u/∂t + ∇ ⋅ S = -J ⋅ E, where u is charge density.
• S = E × H is the Poynting vector.
• H = B/μ and S = (E × B)/μ, where μ= 4π/c.
• The final equation states that: S = (c/4π) E × B.
• S=ck ExB when k = 1/4π, and 1/√(μ ) = c.
• S = (E × B)/μ if c = 1/√(μ ).
• A x B = |A||B| sinθ, and for θ = 90°, sin 90° = 1.
• Electromagnetic waves carry energy as they travel through empty space.
• There is an energy density associated with an electric field E and a magnetic field B.
• Poynting vector is the rate of energy transport per unit area.
• Electric and magnetic fields are related in free space: B=E / c.
• It simply represents the energy flux density in term of electric and magnetic field using the pointing vector.
• Power flow in waveguides and antennas are applications of the Poynting vector.
• Radiation pressure in optics is an application of the Poynting vector.

Conclusion

• Electromagnetic potentials simplify field calculations.
• The Poynting vector explains energy transfer.
• Applications include wireless communication and optics.