Hertzian Dipole Radiation Preview

Physics Electricity and Magnetism • Ac Circuits and Electromagnetic Waves

Written by STEM Calculators Team Published January 18, 2026 Updated February 24, 2026

Preview far-field radiation from a short (Hertzian) dipole: radiated power scaling, wavelength, and the classic sin²θ doughnut pattern.

Inputs accept 1e-3, pi, e, sqrt(2), sin(), cos(), tan(), ln(), log(), abs(). Use * for multiplication.

Amplitude model: Frequency input: f (Hz): ω (rad/s):

Medium extension (εᵣ, μᵣ) + optional “replace μ₀,c” approximation

εᵣ: μᵣ: Use medium in P formula?

Vacuum defaults: εᵣ=1, μᵣ=1. The “medium in P” toggle is a simple approximation for preview purposes.

Electric dipole moment amplitude

p₀ (C·m):

Uses the preview scaling you requested: \(\;P_{\text{avg}}=\dfrac{\mu\,p_0^2\,\omega^4}{12\pi\,c}\;\)

Current moment amplitude (short dipole)

I₀ℓ (A·m):

Common Hertzian-dipole far-field scaling (preview): \(\;P_{\text{avg}}=\dfrac{\mu\,(I_0\ell)^2\,\omega^2}{12\pi\,c}\;\)
Also: \(p_0 \approx \dfrac{I_0\ell}{\omega}\) (useful mapping between models).

Direction + intensity at a distance

Angle θ (deg): Distance r (m): Pattern to plot:

Normalized dipole pattern: \(I(\theta)\propto \sin^2\theta\). Maximum at \(90^\circ\), nulls on axis.

Visual controls

Animation view: Animation speed: Graph text size:

The animation and plot are stacked vertically (as requested).

Ready

Choose inputs and click Compute.

Animation

Emission view shows expanding “pattern rings”. 3D view shows a rotating doughnut-like lobe.

Radiation pattern

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Theory

What is a Hertzian dipole?

A Hertzian (short) dipole is an idealized antenna element with length \(\ell\ll\lambda\), carrying a sinusoidal current. It’s used to understand far-field radiation and angular patterns without the complexity of a real feed and finite geometry.

Validity (preview): \(\ell\ll\lambda\) and observation distance \(r\gg\lambda\).

Radiated power scaling

Two common “amplitude” descriptions are used:

Electric dipole moment \(p(t)=p_0\cos(\omega t)\) (units C·m)
Current moment \(I_0\ell\) (units A·m), often used for short antennas

This calculator provides a preview using these far-field scalings:

\[ P_{\text{avg}}=\frac{\mu\,p_0^2\,\omega^4}{12\pi\,c} \qquad\text{or}\qquad P_{\text{avg}}=\frac{\mu\,(I_0\ell)^2\,\omega^2}{12\pi\,c}. \]

A convenient mapping between them (for sinusoidal excitation) is: \[ p_0 \approx \frac{I_0\ell}{\omega},\qquad I_0\ell\approx p_0\omega. \]

In media, radiation details can be more subtle; the calculator optionally applies a simple approximation by replacing \(\mu_0\to\mu=\mu_0\mu_r\) and \(c\to v=1/\sqrt{\mu\varepsilon}\).

Radiation pattern (doughnut)

The classic Hertzian dipole pattern has nulls along the dipole axis and a maximum in the plane perpendicular to the dipole:

\[ I(\theta)\propto \sin^2\theta. \]

If the total radiated power is \(P_{\text{avg}}\), the power per solid angle can be written as \[ U(\theta)=\frac{3P_{\text{avg}}}{8\pi}\sin^2\theta, \] and the intensity (power per area) at distance \(r\) is \[ S(\theta)=\frac{U(\theta)}{r^2}=\frac{3P_{\text{avg}}}{8\pi r^2}\sin^2\theta. \]

Maximum: \(\theta=90^\circ\). Nulls: \(\theta=0^\circ,180^\circ\).

Unification timeline (website tip)

Faraday (1830s): induction and changing magnetic flux
Maxwell (1860s): displacement current → consistent continuity + EM waves
Hertz (1880s): radio waves observed → dipole radiation becomes practical

Tip: on your page, you can visually connect this tool to the Poynting vector tool (energy flow) and EM wave properties (speed, wavelength).