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Antenna Theory
In antenna theory, the isotropic radiator is a theoretical radiator having a directivity of 0 dBi (dB relative to isotropic), which means that the radiator equally transmits (or receives) electromagnetic radiation to/from any arbitrary direction.

In reality, a coherent isotropic radiator cannot exist, as the isotropic radiator, with a radiation pattern (as expressed in spherical coordinates) of
 * $${\vec E}\left(r,\theta,\phi\right)=\frac{e^{-jkr}}{4\pi r}{\hat u}\left(\theta,\phi\right)$$
 * (note that the magnitude of this function is independent of the spherical angles $$\displaystyle\theta$$ and $$\displaystyle\phi$$, but it is permissible for the vector's direction, as represented by the unit vector $${\hat u}$$ to be a function of $$\displaystyle\theta$$ and $$\displaystyle\phi$$)

would violate the Helmholtz Wave Equation, as derived from Maxwell's Equations.

Although the Sun and other stars radiate equally in all directions, their radiation pattern does not violate Maxwell's equations, because radiation from a star is incoherent. Sound waves can also expand uniformly in all directions, but sound waves are longitudinal waves and not transverse waves.

Even though an isotropic radiator cannot exist in practice, antenna directivity is usually compared to the directivity of an isotropic radiator, because the gain (which is closely related to directivity) relative to an isotropic radiator is useful in the Friis transmission equation. The smallest directivity a radiator can have relative to an isotropic radiator, is a Hertzian Dipole, which has 1.76 dBi.

Hairy ball theorem
Another way to explain why an isotropic radiator cannot exist is by using the hairy ball theorem, which asserts that a continuous vector field tangent to the surface of the sphere, must fall to zero at at least one point on the sphere. This means that there is some direction for which the electric field must be zero, and hence, non-uniform.

Quasi-ideal isotropic receiver
In EMF measurements applications, an isotropic receiver (also called isotropic antenna), is a field measurement instrument which allows to obtain the total field independently of the tri-axial orthogonal arrangement chosen for orientation of the device itself.

In practice a quasi-ideal isotropic receiver is obtained with three orthogonal sensing devices with a radiation diagram of the omnidirectional type $$ \sin (\theta)$$, like that of short dipole and small loop antennas.

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