We present magnetodielectric measurements in single crystals of the cubic spin-1/2 compound Cu$_2$OSeO$_3$. A magnetic field-induced electric polarization ($vec{P}$) and a finite magnetocapacitance (MC) is observed at the onset of the magnetically ordered state ($T_c = 59$ K). Both $vec{P}$ and MC are explored in considerable detail as a function of temperature (T), applied field $vec{H}_a$, and relative field orientations with respect to the crystallographic axes. The magnetodielectric data show a number of anomalies which signal magnetic phase transitions, and allow to map out the phase diagram of the system in the $H_a$-T plane. Below the 3up-1down collinear ferrimagnetic phase, we find two additional magnetic phases. We demonstrate that these are related to the field-driven evolution of a long-period helical phase, which is stabilized by the chiral Dzyalozinskii-Moriya term $D vec{M} cdot(bs{ abla}timesvec{M})$ that is present in this non-centrosymmetric compound. We also present a phenomenological Landau-Ginzburg theory for the ME$_H$ effect, which is in excellent agreement with experimental data, and shows three novel features: (i) the polarization $vec{P}$ has a uniform as well as a long-wavelength spatial component that is given by the pitch of the magnetic helices, (ii) the uniform component of $vec{P}$ points along the vector $(H^yH^z, H^zH^x, H^xH^y)$, and (iii) its strength is proportional to $eta_parallel^2-eta_perp^2/2$, where $eta_parallel$ is the longitudinal and $eta_perp$ is the transverse (and spiraling) component of the magnetic ordering. Hence, the field dependence of P provides a clear signature of the evolution of a conical helix under a magnetic field. A similar phenomenological theory is discussed for the MC.
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