Electric Dipole Radiation and Antennas
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You should be comfortable with retarded potentials, the definition of the electric dipole moment, and basic vector calculus in spherical coordinates. Familiarity with the Larmor formula for a single charge is helpful but not strictly required.
Why this matters
The oscillating electric dipole is the fundamental building block of all radio transmission.
While the Larmor formula describes radiation from a single point charge, real-world antennas consist of many charges moving in concert. The electric dipole approximation captures the leading-order radiation from any antenna that is small compared to the wavelength. Understanding this case explains why AM radio towers are tall, why Wi-Fi antennas have specific shapes, and how energy flows from a transmitter into the vacuum. It bridges the gap between microscopic particle dynamics and macroscopic engineering.
The Physical Picture
Imagine a tiny antenna where positive and negative charges oscillate back and forth along a wire. At any instant, the separation between the positive and negative charge clouds defines an electric dipole moment. When this separation changes with time, the electric field lines cannot adjust instantaneously everywhere in space. Instead, a disturbance propagates outward at the speed of light. Far from the antenna, this disturbance looks like a wave. The key insight is that the radiation pattern is not uniform; it is strongest perpendicular to the axis of oscillation and zero along the axis. This "doughnut" shape is a direct consequence of the vector nature of the electric field and the geometry of the dipole.
Radiation Zone Fields
The derivation card below establishes the fields in the radiation zone, where the distance \( r \) is much larger than the wavelength \( \lambda \) and the antenna size. In this region, the fields simplify dramatically. The electric field is purely transverse, pointing in the \( \hat{\boldsymbol\theta} \) direction, and falls off as \( 1/r \). This is distinct from the static dipole field, which falls off as \( 1/r^3 \). The magnetic field is related to the electric field by the right-hand rule and the speed of light, ensuring the wave propagates radially outward.
Notice the dependence on \( \sin\theta \). This factor dictates the angular distribution of the radiation. The term \( \cos[\omega(t-r/c)] \) represents the oscillating nature of the wave, with the phase delay \( r/c \) accounting for the finite travel time of the signal.
Angular Power Distribution
The power radiated per unit solid angle is proportional to the square of the electric field amplitude. Since the field amplitude contains \( \sin\theta \), the power distribution scales with \( \sin^2\theta \). This means the antenna radiates most strongly in the plane perpendicular to its axis and nothing along the axis. This is a crucial design consideration: if you place a receiver directly above a vertical dipole, you will detect no signal, regardless of distance.
The \( \omega^4 \) dependence is significant. It implies that doubling the frequency increases the radiated power by a factor of 16, assuming the dipole moment amplitude remains constant. This strong frequency dependence is why high-frequency antennas can be smaller yet radiate efficiently, and why low-frequency AM radio requires very large structures to achieve comparable radiation efficiency.
Total Radiated Power
To find the total power, we integrate the angular distribution over the entire sphere. The integral of \( \sin^2\theta \) over solid angle is \( 8\pi/3 \). Multiplying this by the coefficient in the differential power expression yields the total time-averaged radiated power. This result is the dipole analogue of the Larmor formula, scaled by the dipole moment and frequency.
This formula allows us to calculate the energy loss of an oscillating dipole. It also serves as a benchmark for antenna efficiency. If an antenna is driven by an AC source, the power supplied by the source must match this radiated power plus any resistive losses in the conductor.
The derivation
Electric Dipole Radiation
For an oscillating point electric dipole \( \vec p(t) = p_0 \cos(\omega t)\,\hat{\mathbf z} \) in vacuum, we derive the radiation-zone fields, the time-averaged angular power distribution \( \dfrac{d\langle P\rangle}{d\Omega} = \dfrac{\mu_0 p_0^2 \omega^4}{32\pi^2 c}\,\sin^2\theta \), and the total time-averaged radiated power \( \langle P\rangle = \dfrac{\mu_0 p_0^2 \omega^4}{12\pi c} \), starting from the retarded potentials and working consistently to leading order in \( d/\lambda \), \( d/r \), and \( \lambda/r \).
A common misconception is that the \( 1/r \) field is the only field present. In reality, the full field contains \( 1/r^2 \) (induction) and \( 1/r^3 \) (static) terms. In the radiation zone, these are negligible, but near the antenna, they dominate. Confusing these regimes leads to incorrect calculations of near-field energy storage and reactive power. C
Worked example
Problem. A short dipole antenna has a dipole moment amplitude \( p_0 = 1.0 \times 10^{-9} \ \mathrm{C \cdot m} \) and oscillates at a frequency \( f = 100 \ \mathrm{MHz} \). Calculate the total time-averaged radiated power. Use \( \varepsilon_0 = 8.85 \times 10^{-12} \ \mathrm{F/m} \) and \( c = 3.0 \times 10^8 \ \mathrm{m/s} \).
- Determine the angular frequency: \( \omega = 2\pi f = 2\pi (100 \times 10^6) \ \mathrm{rad/s} \approx 6.28 \times 10^8 \ \mathrm{rad/s} \).
- Calculate \( \omega^4 \): \( (6.28 \times 10^8)^4 \approx 1.55 \times 10^{35} \ \mathrm{rad^4/s^4} \).
- Calculate \( p_0^2 \): \( (1.0 \times 10^{-9})^2 = 1.0 \times 10^{-18} \ \mathrm{C^2 \cdot m^2} \).
- Compute the denominator: \( 12\pi \varepsilon_0 c^3 = 12\pi (8.85 \times 10^{-12}) (3.0 \times 10^8)^3 \approx 12\pi (8.85 \times 10^{-12}) (2.7 \times 10^{25}) \approx 9.01 \times 10^{15} \).
- Calculate \( \langle P\rangle \): \( \frac{(1.0 \times 10^{-18})(1.55 \times 10^{35})}{9.01 \times 10^{15}} \approx \frac{1.55 \times 10^{17}}{9.01 \times 10^{15}} \approx 17.3 \ \mathrm{W} \).
Units check. \(\varepsilon_0 c^3\) has units \(\mathrm{C^2\,N^{-1}\,m^{-2}} \times \mathrm{m^3\,s^{-3}} = \mathrm{C^2\,m\,N^{-1}\,s^{-3}}\), and the numerator \(p_0^2\omega^4\) has units \(\mathrm{C^2\,m^2\,s^{-4}}\). The ratio is \(\mathrm{N\,m\,s^{-1}} = \mathrm{J\,s^{-1}} = \mathrm{W}\).
Because \(\langle P\rangle \propto \omega^4\), the same dipole moment driven at 1 MHz (AM band) would radiate \(10^{8}\) times less, only about \(0.17\ \mu\mathrm{W}\). This is why low-frequency transmitters need very tall antennas to reach a useful dipole moment.
Three quick tests
- What is the angular dependence of the radiated power from an electric dipole?
Answer
It scales as \( \sin^2\theta \), where \( \theta \) is the angle from the dipole axis. - How does the total radiated power scale with frequency?
Answer
It scales as \( \omega^4 \), or \( f^4 \). - In which direction is the radiation intensity zero?
Answer
Along the axis of the dipole (where \( \theta = 0 \) or \( \pi \)).
Problem set for PU-204 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes checked line by line by the editor; every number in the worked example was recomputed.