In Fig, 25-31, a 20.0 V battery is connected across...

In Fig, $25-31,$ a 20.0 $\mathrm{V}$ battery is connected across capacitors of capacitances $C_{1}=C_{6}=3.00 \mu \mathrm{F}$ and $C_{3}=C_{5}=2.00 C_{2}=2.00 C_{4}=4.00 \mu \mathrm{F}$ . What are (a) the equivalent capacitance $C_{\mathrm{eq}}$ of the capacitors and (b) the charge stored by $C_{\mathrm{eq}} ?$ What are $(\mathrm{c}) V_{1}$ and $(\mathrm{d}) q_{1}$ of capacitor $1,$ (e) $V_{2}$ and $(\mathrm{f}) q_{2}$ of capacitor $2,$ and $(\mathrm{g}) V_{3}$ and (h) $q_{3}$ of capacitor 3$?$

In Fig, $25-31,$ a 20.0 $\mathrm{V}$ battery is connected across capacitors
of capacitances $C_{1}=C_{6}=3.00 \mu \mathrm{F}$ and $C_{3}=C_{5}=2.00 C_{2}=2.00 C_{4}=4.00 \mu \mathrm{F}$ . What are (a) the equivalent capacitance $C_{\mathrm{eq}}$ of the capacitors and (b) the charge stored by $C_{\mathrm{eq}} ?$ What are $(\mathrm{c}) V_{1}$ and $(\mathrm{d}) q_{1}$ of capacitor $1,$ (e) $V_{2}$ and $(\mathrm{f}) q_{2}$ of capacitor $2,$ and $(\mathrm{g}) V_{3}$ and (h) $q_{3}$ of capacitor 3$?$

Key Concepts

Series Connection

In a series connection of capacitors, each capacitor carries the same charge, and the overall equivalent capacitance is found by taking the reciprocal of the sum of the reciprocals of the individual capacitances. This property is crucial when determining how the total applied voltage is distributed among the capacitors in series.

Parallel Connection

When capacitors are connected in parallel, they all have the same voltage across them since both of their terminals are connected to common nodes. The equivalent capacitance in parallel is simply the sum of the individual capacitances, directly affecting the total charge stored in that section of the circuit.

Equivalent Capacitance Calculation

The concept of equivalent capacitance involves reducing complex capacitor networks—often made up of combinations of series and parallel connections—into a single capacitor that has the same effect on the circuit as the original network. This simplification is key to analyzing overall circuit behavior, such as determining the total charge stored when a voltage is applied.

Charge-Voltage Relationship in Capacitors

The charge stored on a capacitor is directly proportional to both its capacitance and the voltage across it, expressed mathematically as q = CV. This fundamental relationship is used to calculate either the stored charge or the voltage across an individual capacitor once the other quantities are known.

Battery Voltage and Voltage Distribution

When a battery is connected to a capacitor network, the total voltage supplied by the battery is divided among the capacitor elements. In series arrangements, this voltage division follows the inverse proportionality to capacitance, while in parallel arrangements, every branch experiences the full battery voltage. Understanding how the battery voltage distributes itself across the network is essential for analyzing both the voltage across and charge on each capacitor.

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