Maxwell's Equations and Displacement Current

Vector Operators

• The operator ∇ is called "del" and is not a vector, but behaves like a vector in terms of dot product and cross product.

Divergence of a Vector

• The divergence of a vector field at a point represents the flux generation per unit volume centered at that point.
• The divergence of a vector field is defined as: ∇⋅A = (∂Ax/∂x) + (∂Ay/∂y) + (∂Az/∂z)
• The divergence of a vector field can be positive, negative, or zero, indicating source, sink, or no flux at a point.

Curl of a Vector

• The curl of a vector is an axial (rotational) vector whose magnitude is the maximum circulation of A per unit area.
• The curl of a vector is defined as: ∇×A = (∂Az/∂y) - (∂Ay/∂z) i + (∂Ax/∂z) - (∂Az/∂x) j + (∂Ay/∂x) - (∂Ax/∂y) k
• The curl of a vector represents the rotation of the vector field.

Ampere's Law

• Ampere's law states that the circulation of the magnetic field around a closed loop is proportional to the current flowing through the loop.
• The law is limited to magnetostatics and only applies to steady currents.
• Maxwell's correction to Ampere's law introduces the concept of displacement current, which allows for time-varying electric fields.

Maxwell's Equations

• Maxwell's equations are a set of four equations that describe the behavior of electric and magnetic fields.
• The equations are:
  • Gauss's law: ∇⋅E = ρ/ε₀
  • Gauss's law for magnetism: ∇⋅B = 0
  • Faraday's law of induction: ∇×E = -∂B/∂t
  • Ampere's law with Maxwell's correction: ∇×B = μ₀J + μ₀ε₀∂E/∂t
• These equations form the basis of classical electromagnetism.

Maxwell's Equations in Vacuum

• In vacuum, the Maxwell's equations take the form:
  • ∇⋅E = 0
  • ∇⋅B = 0
  • ∇×E = -∂B/∂t
  • ∇×B = μ₀ε₀∂E/∂t
• The solutions to these equations describe the electric and magnetic fields in free space.

Solution of Maxwell's Equations in Vacuum

• The solution to the Maxwell's equations in vacuum involves the electric and magnetic fields, E and B, which are perpendicular to each other and to the direction of propagation.
• The wave impedance in free space is Z₀ = √(μ₀/ε₀) ≈ 377 Ω.

Maxwell's Equations in a Conducting Medium

• In a conducting medium, the Maxwell's equations take the form:
  • ∇⋅E = ρ/ε
  • ∇⋅B = 0
  • ∇×E = -∂B/∂t
  • ∇×B = μJ + με∂E/∂t
• The solutions to these equations describe the electric and magnetic fields in a conducting medium.

Solution of Maxwell's Equations in a Conducting Medium

• The solution to the Maxwell's equations in a conducting medium involves the electric and magnetic fields, E and B, which are not in phase.
• The wave vector k is a complex quantity, indicating a phase lag between E and B.
• The magnitude of the complex wave vector can be written as k = k+ + ik-, where k+ and k- are real and imaginary parts of the wave vector.

Consequence of Complex Wave Vector in Conducting Medium

• The complex wave vector indicates a phase lag between the electric and magnetic fields in a conducting medium.
• For good conductors, the phase lag is significant, while for poor conductors, the phase lag is negligible.
• The complex wave vector has important implications for the behavior of electromagnetic waves in conducting media.