# How to create time variable magnetic fields with high spatial resolution?

Arbitrary shapes of magnetic fields are not possible, but divergenceless magnetic fields are physical. Various shapes of magnetic fields can be created by passing various, time dependent currents through a group of adjacent solenoids, but the use of solenoids imposes a significant limitation on spatial resolution of the created magnetic field.

Is there a way to generate time variable magnetic fields with high spatial resolution? Can it be done without solenoids? Without moving parts?

The difficulty is creating high spatial frequency components in the field variation at positions that are significantly removed from the sources. This difficulty arises through the phenomenon of evanescence. The phenomenon is best demonstrated by a simple example. Suppose you have a magnetic field that is time / space varying on the plane at $z=0$ bounding a half-infinite region. At $z=0$, the magnetic field variation is of the form (perhaps set up by a system of sheet currents, or effective sheet currents modelling the aperture of a horn antenna) $\vec{H}(0,\,t) = H_0\,\exp(i\,(k_s\,x-\omega\,t))\,\hat{y}$, where the spatial frequency $k_s$ is much greater than the freespace wavelength at the frequency $\omega$. If you solve Maxwell's equations for this boundary value problem, you find that the axial component of the wavevector is large and imaginary, given by $i\,\sqrt{k_s^2-k^2}$, where $k=c\,\omega$ is the freespace wavenumber at the frequency $\omega$. This corresponds to a field that is heavily attenuated with increasing distance from the source currents. Thus, only spatial frequencies of the order of the freespace wavenumber or less can be "projected" onto regions far removed from the sources.