  We present a new algorithm for computing the electromagnetic fields of
currents inside and outside of finite current sources, for arbitrary time
variations in the currents. Unexpectedly, we find that our solutions for these
fields are free of the concepts of differential calculus, in that our solutions
only involve the currents and their time integrals, and do not involve the time
derivatives of the currents. As examples, we give the solutions for two
configurations of current: a planar solenoid and a rotating spherical shell
carrying a uniform charge density. For slow time variations in the currents, we
show that our general solutions reduce to the standard expressions for the
fields in classic magnetic dipole radiation. In the limit of extremely fast
turn-on of the currents, we show that for our general solutions the amount of
energy radiated is exactly equal to the magnetic energy stored in the static
fields a long time after current creation. We give three associated problem
statements which can be used in courses at the undergraduate level, and one
problem statement suitable for courses at the graduate level. These problems
are of physical interest because: (1) they show that current systems of finite
extent can radiate even during time intervals when the currents are constant;
(2) they explicitly display transit time delays across a source associated with
its finite dimensions; and (3) they allow students to see directly the origin
of the reaction forces for time-varying systems
