Resistors and Capacitors in Series and Parallel (4C) - MCAT Chemical and Physical Foundations of Biological Systems

$V_C(t)=V_0e^{-t/\tau}$. During discharge, capacitor voltage decays exponentially from its initial value over time constant τ.

$I(t)=\frac{V}{R}e^{-t/\tau}$. Charging current starts at maximum V/R and decays exponentially as the capacitor voltage rises.

$I(t)=\frac{V_0}{R}e^{-t/\tau}$. Discharging current begins at V₀/R and decreases exponentially as stored charge depletes.

$V=8\ \text{V}$. Voltage divides proportionally; the 4Ω resistor takes 4/6 of total voltage.

$C_{eq}=\frac{6}{5}\ \mu\text{F}$. For two series capacitors, equivalent capacitance is their product divided by their sum.

$C_{eq}=5\ \mu\text{F}$. Parallel capacitances sum directly to yield the equivalent capacitance.

$R_{eq}=\frac{6}{5}\ \Omega$. For two parallel resistors, equivalent resistance is their product divided by their sum.

$R_{eq}=5\ \Omega$. Series resistances add directly to give the total equivalent resistance.

$\frac{V_{2\mu F}}{V_{4\mu F}}=2$. With equal charge, voltage ratio is inverse to capacitance ratio, so V_{2μF} is twice V_{4μF}.

$\frac{1}{C_{eq}}=\frac{1}{C_1}+\frac{1}{C_2}$. For series capacitors, reciprocals add since charges are equal and total voltage is the sum of individual voltages.

Current $I$ is the same through all series resistors. In a series circuit, current has only one path, remaining constant through each resistor by conservation of charge.

$R_{eq}=\frac{R}{2}$. Two identical parallel resistors double the conductance, halving the equivalent resistance.

Voltage $V$ is the same across all parallel branches. Parallel branches connect across the same points, experiencing identical potential difference by definition.

Charge $Q$ is the same on all series capacitors. Series capacitors share the same current, resulting in equal charge accumulation on each plate.

Voltage $V$ is the same across all parallel capacitors. Parallel capacitors are connected across the same potential difference, equalizing voltage on each.

$V_i=V_{tot}\frac{R_i}{\sum R}$. Voltage drops proportionally to each resistor's share of the total resistance in a series circuit.

$V_i=\frac{Q}{C_i}$. With constant charge on series capacitors, voltage on each is inversely proportional to its capacitance.

$I_{tot}=\sum I_i$. By Kirchhoff's current law, total current entering a parallel junction equals the sum of branch currents.

$Q_{tot}=\sum Q_i$. With identical voltage, total charge on parallel capacitors sums the charges on each.

$\tau=R_{eq}C_{eq}$. The time constant in RC circuits is the product of equivalent resistance and capacitance, governing exponential behavior.

$V_C(t)=V\left(1-e^{-t/\tau}\right)$. Capacitor voltage approaches the battery voltage exponentially during charging in an RC circuit.

$C_{eq}=\frac{C}{2}$. Two identical series capacitors sum reciprocals, yielding half the individual capacitance.

$R_{eq}=R_1+R_2$. Resistances in series add directly as current flows through each sequentially, increasing total opposition to flow.

$C_{eq}=C_1+C_2$. Capacitances in parallel add because they share the same voltage, and total charge is the sum of individual charges.