charge does not radiate (emit electromagnetic radiation). This term requires additionally that a component of the charge acceleration be in a direction transverse to the line which connects the charge and the observer of the field . The direction of the field associated with this radiative term is toward the fully time-retarded position of the charge (i.e. where the charge was when it was accelerated). Derivation The scalar and vector potentials satisfy the nonhomogeneous electromagnetic wave equation where the sources are expressed with the charge and current densities and and the Ampère-Maxwell law is: Since the potentials are not unique, but have gauge freedom, these equations can be simplified by gauge fixing. A common choice is the Lorenz gauge condition: Then the nonhomogeneous wave equations become uncoupled and symmetric in the potentials: Generally, the retarded solutions for the scalar and vector potentials (SI units) are and where is the retarded time and and satisfy the homogeneous wave equation with no sources and the boundary conditions. In the case that there are no boundaries surrounding the sources then and . For a moving point charge whose trajectory is given as a function of time by , the charge and current densities are as follows: where is the three-dimensional Dirac delta function and is the velocity of the point charge. Substituting into the expressions for the potential gives These integrals are difficult to evaluate in their present form, so we will rewrite them by replacing with and integrating over the delta distribution : We exchange the order of integration: The delta function picks out which allows us to perform the inner integration with ease. Note that is a function of , so this integration also fixes . The retarded time is a function of the field point and the source trajectory , and hence depends on . To evaluate this integral, therefore, we need the identity where each is a zero of . Because there is only one retarded time for any given space-time coordinates and source trajectory , this reduces to: where and are evaluated at the retarded time , and we have used the identity with . Notice that the retarded time is the solution of the equation . Finally, the delta function picks out , and which are the Liénard–Wiechert potentials. Lorenz gauge, electric and magnetic fields In order to calculate the derivatives of and it is convenient to