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IB DP Physics: HL復習筆記11.3.3 Capacitors in Series & Parallel

Category: IB課程, 教材筆記, 福利干貨 Date: 2022年8月2日下午12:30

Capacitors in Series & Parallel

Capacitors can be combined in series and parallel circuits
The combined?capacitance?depends on whether the capacitors are connected in series or parallel

Capacitors in Series

Consider two parallel plate capacitors?C1?and?C2?connected in series, with a?potential difference? (p.d)?V?across them

19.1-Capacitors-in-series

Capacitors connected in series have different p.d across them but have the same charge

In a series circuit, p.d is?shared?between all the components in the circuit
- Therefore, if the capacitors store the same charge on their plates but have different p.ds, the p.d across?C1?is?V1?and across?C2?is?V2
The total potential difference?V?is the sum of?V1?and?V2

V = V1?+ V2

Rearranging the capacitance equation for the p.d?V?means?V1?and?V2?can be written as:

3.-Derivation-of-C-QV-equation-1

Where the total p.d?V?is defined by the total capacitance

3.-Derivation-of-C-QV-equation-2

Substituting these into the equation V = V1?+ V2?equals:

3.-Derivation-of-C-QV-equation-3

Since the current is the?same?through all components in a series circuit, the charge?Q?is the same through each capacitor and cancels out
Therefore, the equation for combined capacitance of capacitors in?series?is:

3.-Derivation-of-C-QV-equation-4

Capacitors in Parallel

Consider two parallel plate capacitors?C1?and?C2?connected in parallel, each with p.d?V

19.1-Capacitors-in-parallel

Capacitors connected in parallel have the same p.d across them, but different charge

Since the current is?split?across each junction in a parallel circuit, the charge stored on each capacitor is?different
Therefore, the charge on capacitor?C1?is?Q1?and on?C2?is?Q2
The total charge?Q?is the sum of?Q1?and?Q2

Q = Q1?+ Q2

Rearranging the capacitance equation for the charge?Q?means?Q1?and?Q2?can be written as:

Q1?= C1V ??? and????? Q2?= C2V

Where the total charge?Q?is defined by the total capacitance:

Q = CtotalV

Substituting these into the Q = Q1?+ Q2?equals:

CtotalV = C1V + C2V = (C1?+ C2) V

Since the p.d is the?same?through all components in each branch of a parallel circuit, the p.d?V?cancels out
Therefore, the equation for combined capacitance of capacitors in?parallel?is:

Ctotal?= C1?+ C2?+ C3?...

Worked Example

Three capacitors with a capacitance of 23 μF, 35 μF and 40 μF are connected as shown below.

19.1-WE-Capacitors-in-Series-Parallel-question-image

Calculate the total capacitance between points?A?and?B.

Step 1: Calculate the combined capacitance of the two capacitors in parallel

Capacitors in parallel: Ctotal?= C1?+ C2?+ C3?…

Cparallel?= 23 + 35 = 58 μF

Step 2: Connect this combined capacitance with the final capacitor in series

4.-Capacitors-in-Series-Parallel-Worked-Example-equation-1

4.-Capacitors-in-Series-Parallel-Worked-Example-equation-2

Step 3: Rearrange for the total capacitance

4.-Capacitors-in-Series-Parallel-Worked-Example-equation-3

Exam Tip

You will be expected to remember these derivations for your exam, therefore, make sure you understand each step. You should especially make sure to revise how the current and potential difference varies in a series and parallel circuit.

Both the combined capacitance equations look similar to the equations for combined resistance in series and parallel circuits. However, take note that they are the?opposite way?around to each other!

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