(1952) |

where is the energy of the charge. The Poynting vector

(1953) |

where use has been made of [see Equation (1916)], represents the energy flux per unit actual time, . Thus, the energy loss rate of the charge into a given element of solid angle is

where use has been made of Equation (1902). Here, and are angular coordinates used to locate the element of solid angle. It follows from Equation (1922) that

(1955) |

Consider the special case in which the direction of acceleration coincides with the direction of motion. Let us define spherical polar coordinates whose axis points along this common direction. It is easily demonstrated that, in this case, the previous expression reduces to

In the non-relativistic limit, , the radiation pattern has the familiar dependence of dipole radiation. In particular, the pattern is symmetric in the forward ( ) and backward ( ) directions. However, as , the radiation pattern becomes more and more concentrated in the forward direction. The angle for which the intensity is a maximum is

(1957) |

This expression yields as , and as . Thus, for a highly relativistic charge, the radiation is emitted in a narrow cone whose axis is aligned along the direction of motion. In this case, the angular distribution (1958) reduces to

The total power radiated by the charge is obtained by integrating Equation (1958) over all solid angles. We obtain

(1959) |

It is easily verified that

(1960) |

Hence,

(1961) |

which agrees with Equation (1946), provided that .