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UY1: Capacitors In Series And In Parallel

Derive equivalent capacitance formulas for series and parallel capacitor combinations with charge-voltage reasoning.

  • University Physics Year 1
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Learning objectives

  • Analyse capacitance, resistance, energy transfer, and transient circuit behaviour.
Why this matters + quick links

This page gives the UY1 working model/result for Capacitors In Series And In Parallel. You reuse it when you build fields/potentials by symmetry or superposition, and when you connect fields to forces, energy, and circuits.

Equivalent capacitance helps you replace a network with one capacitor that has the same terminal behavior.

1) At a glance

  • Prerequisites: Q = CΔ V, Kirchhoff-style conservation ideas
  • Outcomes: derive and apply series/parallel formulas
  • Key results: 1/C_eq = ∑ᵢ1/Cᵢ (series) C_eq = ∑ᵢ Cᵢ (parallel)
  • Common trap: swapping “same charge” and “same voltage” rules

Motivation / intuition

Series and parallel rules come from simple constraints: in a single series path, charge has nowhere else to go (same Q), while in parallel branches, both elements share the same two nodes (same Δ V). Once you lock those constraints, the equivalent formulas fall out in one line.

2) Setup

Assume ideal capacitors and steady state with a DC source.

  • Series: same charge magnitude on each capacitor
  • Parallel: same potential difference across each capacitor

3) Core derivation/explanation

A) Series connection

For two capacitors: Q₁ = Q₂ = Q Δ V = Δ V₁ + Δ V₂ = Q/C₁ + Q/C₂ So, 1/C_eq = (Δ V)/Q = 1/C₁ + 1/C₂ Generalizes to sums of reciprocals.

B) Parallel connection

For two capacitors: Δ V₁ = Δ V₂ = Δ V Q = Q₁ + Q₂ = C₁Δ V + C₂Δ V Hence, C_eq = Q/(Δ V) = C₁ + C₂ Generalizes to direct sum of capacitances.

4) Worked example(s)

Given C₁ = 6 μF and C₂ = 3 μF.

Series: 1/C_eq = 1/6 + 1/3 = 1/2 (μF⁻¹) C_eq = 2 μF

Parallel: C_eq = 6 + 3 = 9 μF

Sanity check: series gives smaller than smallest, parallel gives larger than largest.

Quick checks (units + limits/sign)
  • Units: all C values (including C_eq) must stay in farads; check that you don’t mix μF and F mid-problem.
  • Limits/signs: for series, C_eq must be smaller than the smallest capacitor; for parallel, C_eq must be larger than the largest.

5) Practice set (with hints + answers)

  1. Three capacitors 2,2,2 μF in series: C_eq?
  2. Same three in parallel: C_eq?
  3. In a series branch, which quantity is the same on each capacitor: Q or Δ V?

Hints

  1. Add reciprocals.
  2. Add directly.
  3. Think charge continuity through one path.

Answers

  1. 2/3 μF.
  2. 6 μF.
  3. Q is the same (magnitude).

6) Summary + next steps

  • Series: same charge, voltages split, reciprocal rule.
  • Parallel: same voltage, charges add, direct-sum rule.
  • Quick magnitude checks prevent many algebra mistakes.

Next: Energy Stored In Capacitors Previous: Capacitors And Capacitance Back To Electromagnetism (UY1)