Summary: The time-dependent flow driven by electromagnetic forcing of an electrolytic fluid in the gap of a concentric spheres set-up is studied experimentally and theoretically. The driving Lorentz force is generated by the interaction of an alternating current radially injected through electrodes located at the equatorial zone of the spheres and a dipolar magnetic field produced by a permanent magnet inside the inner sphere. Experimentally, the time-dependent flows were explored in the laminar regime with a Reynolds number \(Re = 640\) and different forcing frequencies, which resulted in oscillatory Reynolds numbers ranging from \(28\) to \(2820\). Velocity profiles in the equatorial line between spheres were obtained with particle image velocimetry. Given the symmetry of the problem at the equatorial plane, asymptotic and approximate solutions for the azimuthal velocity are obtained for the limiting cases of low-\(Re_\omega\) (in real arguments) and high-\(Re_\omega\) (in complex arguments). Furthermore, a general methodology is proposed in such a way that an exact solution for the problem is obtained. The analytical solutions reproduce the main characteristic behaviour of the flow. An estimation of the oscillatory boundary layer due to the electromagnetic forcing is obtained through the exact solution. A full three-dimensional numerical model, that introduces the dipolar magnetic field and the radial dependency of the applied current, is able to quantitatively reproduce both the analytical solutions and the experimental measurements. Additionally, numerical results show a resonant behaviour of the flow when the forcing frequency is approximately \(Re_\omega \approx 560\).