Resistor-Capacitor (RC) Circuits

A collection of capacitors can be replaced at its external terminals by an equivalent capacitance. In parallel, the capacitor voltages are equal and stored plate-charge magnitudes add: $C_{\mathrm{eq},p}=\sum_i C_i$. Adding a parallel capacitor increases the charge that can be stored at the same voltage.

In a series string with initially neutral, isolated intermediate conducting nodes, each capacitor has equal plate-charge magnitude Q. Electrons displaced onto one plate leave an equal deficit on the connected neighboring plate, by conservation of charge. The voltages add: $\Delta V_{\mathrm{total}}=Q\sum_i\frac1{C_i}$, so $\frac1{C_{\mathrm{eq},s}}=\sum_i\frac1{C_i}$. The series equivalent is less than the smallest capacitor. Smaller C has larger voltage for the same series charge.

Do not apply the equal-series-charge statement to arbitrary precharged intermediate nodes without checking their net charges. The standard initially uncharged AP connection supplies the conservation condition.

For an initially uncharged capacitor in a simple series resistor–capacitor circuit driven by an ideal battery, VC starts at zero. Immediately after switching, it acts like a wire for the initial-circuit calculation: $I_0=\epsilon/R,\quad Q_0=0,\quad U_0=0$. This does not mean carriers cross the insulating gap; charge flows to one plate and away from the other through the external wires.

As charge builds, VC = Q/C increases, leaving less source voltage across the resistor. The branch current therefore decreases. The stored energy ½CVC² increases. Charge and voltage rise toward limiting values; current falls toward zero; the curves flatten asymptotically. Voltage and current do not generally change linearly in time.

Long after switching, the charging capacitor acts as an open circuit in its branch: $I_f=0,\quad V_{C,f}=\epsilon,\quad Q_f=C\epsilon,\quad U_f=\tfrac12C\epsilon^2$ for this simple series arrangement. In a branching circuit, other conducting branches may still carry current, and final capacitor voltage is its actual node difference, not automatically the battery voltage. A capacitor that was already charged need not act like a wire at the initial instant; begin with its initial voltage.

Disconnect the source and connect a charged capacitor across a resistor. Initially the capacitor still has its previous charge and voltage; those quantities begin to decrease once current flows. In the simple discharge loop, $|I_0|=V_{C,0}/R$, with current driven from the capacitor’s positive plate through the resistor toward its negative plate. This direction reverses relative to the corresponding charging current through the resistor.

During discharge, plate-charge magnitude, capacitor-voltage magnitude, stored energy, and current magnitude all decrease. They approach zero with flattening curves; the energy becomes resistor heating. With a fixed charging-polarity current arrow, discharge current is negative and its numerical value rises toward zero, even though its magnitude decreases.

Long after discharging in this source-free loop, $Q_f=0,\quad V_{C,f}=0,\quad I_f=0,\quad U_f=0$. For charging or discharging at times much longer than the time constant, steady-state endpoint models are accurate approximations. “Fully charged” or “fully discharged” here means close enough to the asymptotic endpoint for the chosen precision.

The time constant is $\tau=R_{\mathrm{eq}}C_{\mathrm{eq}}$, with units Ω·F = s. Use the resistance relevant to the capacitor’s charge or discharge path and the equivalent capacitance of that connection. For a simple series RC loop this is RC. Increasing either R or C stretches the time scale and makes the changes slower.

One time constant after charging from zero, the charge is about 63% of its final value. One time constant after starting a simple discharge, about 37% of the initial charge remains. Because Q = CVC at fixed C, voltage follows the same fractions. The asymptotic steady states are approached for times much greater than τ.

AP uses these qualitative representations, time-constant comparisons, and quantitative initial/final calculations. Exponential transient formulas, logarithmic elapsed-time calculations, and time-dependent mathematical modeling are not required and are not used in this unit’s assessments.