We present an algorithm for computing the electromagnetic fields due to currents inside and outside of finite sources with a high degree of spatial symmetry for arbitrary time-dependent currents. The solutions for these fields do not involve the time derivatives of the currents but involve only the currents and their time integrals. We give solutions for moving planar sheets of charge, and a rotating spherical shell carrying a uniform charge density. We show that the general solutions reduce to the standard expressions for magnetic dipole radiation for slow time variations of the currents. If the currents are turned on very quickly, the general solutions show that the amount of energy radiated equals the magnetic energy stored in the static fields a long time after current creation. We give three problems which can be used in undergraduate courses and one problem suitable for graduate courses. These problems illustrate that because the generation of radiation depends on what has happened in the past, as em of currents can radiate even during time intervals when the currents are constant due to radiation associated with earlier acceleration.