Animated diagram of waves from an isotropic radiator (red dot). As they travel away from the source, the waves decrease in amplitude by the inverse of distance ${\displaystyle 1/r}$ and in power by the inverse square of distance ${\displaystyle 1/r^{2}}$, shown by the declining contrast of the wavefronts. This diagram only shows the waves in one plane through the source; an isotropic source actually radiates in all three dimensions.
A depiction of an isotropic radiator of sound, published in Popular Science Monthly in 1878. Note how the rings are even and of the same width all the way around each circle, though they fade as they move away from the source.

An isotropic radiator is a theoretical point source of electromagnetic or sound waves which radiates the same intensity of radiation in all directions. It has no preferred direction of radiation. It radiates uniformly in all directions over a sphere centred on the source. Isotropic radiators are used as reference radiators with which other sources are compared, for example in determining the gain of antennas. A coherent isotropic radiator of electromagnetic waves is theoretically impossible, but incoherent radiators can be built. An isotropic sound radiator is possible because sound is a longitudinal wave.

Whether a radiator is isotropic is independent of whether it obeys Lambert's law. As radiators, a spherical black body is both, a flat black body is Lambertian but not isotropic, a flat chrome sheet is neither, and by symmetry the Sun is isotropic, but not Lambertian on account of limb darkening.

## Physics

In physics, an isotropic radiator is a point radiation or sound source. At a distance, the sun is an isotropic radiator of electromagnetic radiation.

### Antenna theory

In antenna theory, an isotropic antenna is a hypothetical antenna radiating the same intensity of radio waves in all directions. It thus is said to have a directivity of 0 dBi (dB relative to isotropic) in all directions.

In reality, a coherent isotropic radiator of linear polarization can be shown to be impossible. Its radiation field could not be consistent with the Helmholtz wave equation (derived from Maxwell's equations) in all directions simultaneously. Consider a large sphere surrounding the hypothetical point source, so that at that radius the wave over a reasonable area is essentially planar. The electric (and magnetic) field of a plane wave in free space is always perpendicular to the direction of propagation of the wave. So the electric field would have to be tangent to the surface of the sphere everywhere, and continuous along that surface. However the hairy ball theorem shows that a continuous vector field tangent to the surface of a sphere must fall to zero at one or more points on the sphere, which is inconsistent with the assumption of an isotropic radiator with linear polarization.

Incoherent isotropic radiators are possible and do not violate Maxwell's equations.[citation needed] Acoustic isotropic radiators are possible because sound waves in a gas or liquid are longitudinal waves and not transverse waves.

Even though an isotropic antenna cannot exist in practice, it is used as a base of comparison to calculate the directivity of actual antennas. Antenna gain ${\displaystyle \scriptstyle G}$, which is equal to the antenna's directivity multiplied by the antenna efficiency, is defined as the ratio of the intensity ${\displaystyle \scriptstyle I}$ (power per unit area) of the radio power received at a given distance from the antenna (in the direction of maximum radiation) to the intensity ${\displaystyle \scriptstyle I_{\text{iso}}}$ received from a perfect lossless isotropic antenna at the same distance. This is called isotropic gain

${\displaystyle G={I \over I_{\text{iso}}}\,}$

Gain is often expressed in logarithmic units called decibels (dB). When gain is calculated with respect to an isotropic antenna, these are called decibels isotropic (dBi)

${\displaystyle G\mathrm {(dBi)} =10\log {I \over I_{\text{iso}}}\,}$

The gain of any perfectly efficient antenna averaged over all directions is unity, or 0 dBi.

In EMF measurements applications, an isotropic receiver (also called isotropic antenna) is a calibrated radio receiver with an antenna which approximates an isotropic reception pattern; that is, it has close to equal sensitivity to radio waves from any direction. It is used as a field measurement instrument to measure electromagnetic sources and calibrate antennas. The isotropic receiving antenna is usually approximated by three orthogonal antennas or sensing devices with a radiation pattern of the omnidirectional type ${\displaystyle \sin(\theta )}$, such as short dipoles or small loop antennas.

The parameter used to define accuracy in the measurements is called isotropic deviation.

#### Derivation of isotropic antenna aperture

Imagine two cavities in thermal equilibrium. A lossless antenna in one cavity is connected to a matched impedance inside the second cavity. Using the Rayleigh-Jeans approximation

${\displaystyle B_{\nu }={\frac {2kT}{\lambda ^{2}}},}$

the power received by the antenna over a narrow frequency band is

${\displaystyle P_{\nu }=A_{e}S_{\mathrm {matched} }=A_{e}{\frac {S}{2}}={\frac {1}{2}}\left(\int _{4\pi }A_{e}B_{\nu }d\Omega \right)d\nu .}$

Since the cavities are in thermal equilibrium, this equals the Nyquist spectral power of the resistor

${\displaystyle P_{\nu }=kTd\nu }$

in that same frequency band, thus

${\displaystyle {\frac {1}{2}}\left(\int _{4\pi }A_{e}B_{\nu }d\Omega \right)d\nu =kTd\nu .}$
${\displaystyle {\frac {1}{2}}\left(\int _{4\pi }A_{e}{\frac {2kT}{\lambda ^{2}}}d\Omega \right)d\nu =kTd\nu .}$

Being isotropic, Ae is constant in every direction, thus

${\displaystyle A_{e}{\frac {kT}{\lambda ^{2}}}\int _{4\pi }d\Omega =kT.}$
${\displaystyle A_{e}={\frac {\lambda ^{2}}{4\pi }}.}$[1]

### Optics

In optics, an isotropic radiator is a point source of light. The sun approximates an isotropic radiator of light. Certain munitions such as flares and chaff have isotropic radiator properties.

### Sound

An isotropic radiator is a theoretical perfect speaker exhibiting equal sound volume in all directions.