Jawaharlal Nehru Technological University Hyderabad B.Tech III Year ANTENNAS AND WAVE PROPAGATION Previous Question Paper (February - 2022 - R13)

A small electric dipole, or Hertzian dipole, consists of a short conducting wire of length dl carrying a time-harmonic current I = I₀ cos(ωt). Radiation occurs because the time-varying current creates a time-varying charge distribution at the ends of the wire, which in turn generates time-varying electric and magnetic fields that propagate outward as electromagnetic waves.

The radiation mechanism is governed by Maxwell's equations. As the current oscillates, it produces a magnetic field H in the azimuthal direction and an electric field E with both radial and tangential components. In the far-field (radiation) zone, where the distance r is much greater than the wavelength λ, the fields are dominated by the 1/r terms. The radiated electric field is proportional to the sine of the angle θ from the dipole axis, resulting in a donut-shaped radiation pattern.

The radiated power is determined by the Poynting vector, which represents the directional energy flux density. The time-average power radiated by a short dipole is given by:

This shows that the radiation resistance is proportional to the square of the ratio of the dipole length to the wavelength. Consequently, for efficient radiation, the dipole length must be a significant fraction of the wavelength.

The Helmholtz theorem states that any vector field F(r) that is sufficiently smooth and vanishes at infinity can be uniquely decomposed into the sum of an irrotational (curl-free) component and a solenoidal (divergence-free) component. Mathematically, this is expressed as:

Where Φ is a scalar potential and A is a vector potential. The theorem further specifies that these potentials are determined by the divergence and curl of the field:

• The irrotational part is defined by the divergence of the field: ∇ · F = ρ.
• The solenoidal part is defined by the curl of the field: ∇ × F = J.

In electromagnetics, this theorem provides the theoretical foundation for defining the scalar potential V and the magnetic vector potential A, allowing Maxwell's equations to be solved by decoupling the electric and magnetic field vectors into these simpler potential functions.