Section 4.4: Capacitors
Capacitors are devices that store electrical energy in an electric field. They are widely used in DC and AC circuits for energy storage, filtering, and timing applications.
- Capacitance (C): Ability to store charge, measured in farads (F): \[ C = \frac{Q}{V} \] where \(Q\) = charge in coulombs, \(V\) = voltage across the capacitor.
- Energy Stored: \[ U = \frac{1}{2} C V^2 \]
- Series Capacitors: Total capacitance: \[ \frac{1}{C_\text{total}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n} \]
- Parallel Capacitors: Total capacitance: \[ C_\text{total} = C_1 + C_2 + \dots + C_n \]
Example: Series and Parallel Capacitors
Find the total capacitance of two capacitors C1 = 4 μF and C2 = 6 μF connected (a) in series, (b) in parallel.
(a) Series: \( \frac{1}{C_\text{total}} = \frac{1}{4} + \frac{1}{6} = \frac{5}{12} \Rightarrow C_\text{total} = \frac{12}{5} = 2.4 \, \mu\text{F} \)
(b) Parallel: \( C_\text{total} = C_1 + C_2 = 4 + 6 = 10 \, \mu\text{F} \)
Practice Problems
- Two capacitors, 5 μF and 10 μF, are in series across a 12 V battery. Find the total capacitance and voltage across each capacitor.
- Two capacitors, 3 μF and 6 μF, are in parallel across a 9 V battery. Find the total capacitance and charge on each capacitor.
- A 4 μF capacitor is charged to 12 V. Calculate the energy stored.
- Explain why series capacitors have lower total capacitance than individual capacitors.
- Three capacitors 2 μF, 3 μF, and 6 μF are connected in a combination of series and parallel. Determine the total capacitance if 2 μF and 3 μF are in series and then in parallel with 6 μF.