The Fields of a Linear Current Ramp
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We have seen that when the current given by Eq. ( 2.1) is turned on suddenly, a bubble of flux moves outward leaving the dipole field of a uniformly magnetized sphere in its wake. We can use this result to evaluate the field when the current is turned on gradually. The idea is to regard the gradual increase as a sequence of infinitesimal sudden increases, like small steps in a long staircase. Because Maxwell’s equations are linear, we can combine the individual vector potentials of these current steps to obtain the vector potential for the entire ramp.