An electromagnetic resistivity logging tool comprising: a conductive tool body comprising: a first transmitter comprising: a first transmitter annular cavity that encircles the conductive tool body, a first transmitter surface gap that encircles the conductive tool body and electromagnetically couples the first transmitter annular cavity to an external surface of the conductive tool body, and a first transmitter ring of high magnetic permeability material in the first transmitter annular cavity; a first receiver comprising: a first receiver annular cavity (510, 515), a first receiver surface gap (520, 525) that electromagnetically couples the external surface of the conductive tool body to the first receiver annular cavity (510, 515), and a first high magnetic permeability material (535, 540) in the first receiver annular cavity (510, 515); a first transmitter module coupled to opposite sides of the first transmitter surface gap to induce an electric current around the first transmitter annular cavity; and a first receiver module coupled to opposite sides of the first receiver surface gap (520, 525) to detect currents induced around the first receiver annular cavity (510, 515). a conductive tool body comprising: a first transmitter comprising: a first transmitter annular cavity that encircles the conductive tool body, a first transmitter surface gap that encircles the conductive tool body and electromagnetically couples the first transmitter annular cavity to an external surface of the conductive tool body, and a first transmitter ring of high magnetic permeability material in the first transmitter annular cavity; a first receiver comprising: a first receiver annular cavity (510, 515), a first receiver surface gap (520, 525) that electromagnetically couples the external surface of the conductive tool body to the first receiver annular cavity (510, 515), and a first high magnetic permeability material (535, 540) in the first receiver annular cavity (510, 515); a first transmitter comprising: a first transmitter annular cavity that encircles the conductive tool body, a first transmitter surface gap that encircles the conductive tool body and electromagnetically couples the first transmitter annular cavity to an external surface of the conductive tool body, and a first transmitter ring of high magnetic permeability material in the first transmitter annular cavity; a first transmitter annular cavity that encircles the conductive tool body, a first transmitter surface gap that encircles the conductive tool body and electromagnetically couples the first transmitter annular cavity to an external surface of the conductive tool body, and a first transmitter ring of high magnetic permeability material in the first transmitter annular cavity; a first receiver comprising: a first receiver annular cavity (510, 515), a first receiver surface gap (520, 525) that electromagnetically couples the external surface of the conductive tool body to the first receiver annular cavity (510, 515), and a first high magnetic permeability material (535, 540) in the first receiver annular cavity (510, 515); a first receiver annular cavity (510, 515), a first receiver surface gap (520, 525) that electromagnetically couples the external surface of the conductive tool body to the first receiver annular cavity (510, 515), and a first high magnetic permeability material (535, 540) in the first receiver annular cavity (510, 515); a first transmitter module coupled to opposite sides of the first transmitter surface gap to induce an electric current around the first transmitter annular cavity; and a first receiver module coupled to opposite sides of the first receiver surface gap (520, 525) to detect currents induced around the first receiver annular cavity (510, 515).