PHY C18: Combined Capacitance

On By irfansyahrilIn Physics, Study Notes
  • Formulae for combined capacitance
    • in series
    • in parallel
  • Problem-solving

When multiple capacitors are combined in a circuit, we can treat them as a single equivalent capacitor with a different value of capacitance.

Combined capacitance in Series
Potential difference across combined capacitors is the sum of the potential differences of each capacitor in series:
V = V1 + V2 +…

Charge stored on each plate is of the SAME magnitude (just different signs)
q = q1 = q2 = …

Since V = q/C,
& V = V1 + V2 +…

q/C = q/C1 + q/C2 +…

1/C = 1/C1 + 1/C2 +…

Combined capacitance in parallel
Potential difference across all capacitors is the same:
V = V1 = V2 =…

Total charge stored on all plates is the sum of the charges stored on 1 plate of each capacitor:
q = q1 + q2 + …

Since V = q/C,
q = VC
& q = q1 + q2 +…

VC = VC1 + VC2 +…

C = C1 + C2 +…

Problem solving
Generally, networks of capacitors can be dealt with in a similar way as resistor networks – just remember to use the right formulas for total capacitance!

ComponentSeriesParallel
Capacitors1/C = 1/C1 + 1/C2 +…C = C1 + C2 +…
ResistorsR = R1 + R2 +…1/R = 1/R1 + 1/R2 +…

Common problems include:

  • Calculating the total capacitance of a network/circuit
  • Calculating the maximum/minimum capacitance possible given a set of capacitors
    • To get maximum capacitance: arrange all capacitors in parallel
    • To get minimum capacitance: arrange all capacitors in series

Another common problem is figuring out the maximum potential difference across a capacitor network.
An example:
4 identical capacitors each have a maximum voltage of 6V. Given the arrangement below, what is the maximum safe potential difference across the network?

1. Consider parallel capacitor networks as single capacitors & calculate their capacitance  
2. Calculate the ratio of potential difference across each capacitor
Here, we can just consider C having a p.d. of V
3. Identify which capacitor receives the highest potential difference – this will be your limiting capacitorIn this setup, it would be the capacitor with 3V/2 
4. Let that capacitor receive the given maximum potential difference 3V/2 = 6
5. Calculate the total potential difference V across the whole network V = 9

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