Energy Stored by a Capacitor

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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