What are the charge on and the potential difference across each capacitor in FIGURE P26.57?

Verified step by step guidance

1. Step 1: Analyze the circuit configuration. Capacitor C1 is in series with the combination of capacitors C2 and C3, which are connected in parallel. This means the equivalent capacitance of C2 and C3 must be calculated first.
2. Step 2: Calculate the equivalent capacitance of C2 and C3 using the formula for capacitors in parallel: $C$_eq=C₂+C₃. Substitute the values: $C$_eq=40+60=100μF.
3. Step 3: Calculate the total equivalent capacitance of the circuit. Since C1 is in series with the equivalent capacitance of C2 and C3, use the formula for capacitors in series: $\frac{1}{C}$_total=1C₁+1C_eq. Substitute the values: $\frac{1}{C}$_total=116+1100.
4. Step 4: Determine the charge on C1 using the formula $Q=C$_total×V, where V is the voltage of the battery (12 V). Once the charge on C1 is known, the same charge will be present on the equivalent capacitance of C2 and C3 because capacitors in series share the same charge.
5. Step 5: Calculate the potential difference across each capacitor. For C1, use $V$₁=QC₁. For C2 and C3, first calculate the voltage across the equivalent capacitance using $V$_eq=QC_eq, then split the voltage across C2 and C3 proportionally based on their capacitances.

Key Concepts

Capacitance

Capacitance is the ability of a capacitor to store charge per unit voltage, measured in farads (F). In this circuit, capacitors C1, C2, and C3 have capacitances of 16μF, 40μF, and 60μF, respectively. The total capacitance in a circuit depends on whether the capacitors are arranged in series or parallel, affecting how they share voltage and charge.

Voltage in Capacitors

The voltage across a capacitor is directly related to the charge stored and its capacitance, described by the formula V = Q/C, where V is voltage, Q is charge, and C is capacitance. In this circuit, the potential difference across each capacitor must be calculated based on their arrangement and the total voltage supplied by the battery, which is 12V.

Series and Parallel Capacitor Configurations

Capacitors can be connected in series or parallel, affecting their overall behavior. In series, the total capacitance decreases, and the same charge flows through each capacitor, while the voltage divides among them. In parallel, the total capacitance increases, and each capacitor experiences the same voltage. Understanding this configuration is crucial for determining the charge and voltage across each capacitor in the given circuit.