A test field solution is constructed, describing the electromagnetic field around a Kerr black hole which is drifting in an arbitrary direction. An asymptotically uniform magnetic field is assumed with a general orientation (with respect to the rotation axis). Proper choice of the tetrad of a physically relevant observer is discussed. Lines of force of electric and magnetic intensities are explored, as measured by an observer orbiting around the black hole (above the innermost stable circular orbit, ISCO) or freely falling (below the ISCO) down to the horizon.