A capacitor \(C\) is charged with the external voltage \(U_0\) via a series resistor \(R\). How large is the energy conversion efficiency \(\eta\) for the charging?

The energy conversion efficiency reads \[\eta = \frac{W_C}{W_C + W_R}, \] where \(W_C\) is the stored energy in the capacitor \(C\) and \(W_R\) is the ohmic energy loss over the resistance \(R\). The electrical energies result from the power integrals \[ W_i = \int_0^\infty P_i(t)\, dt = \int_0^\infty U_i(t) I(t)\, dt,\] where \( i = R, C.\)

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