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UY1: Transferring Charge And Energy Between Capacitors

Key idea: Solve capacitor-sharing problems using charge conservation, common final voltage, and energy accounting.

  • 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 Transferring Charge And Energy Between Capacitors. You reuse it when you build fields/potentials by symmetry or superposition, and when you connect fields to forces, energy, and circuits.

1) At a glance

  • Prerequisites: parallel rule + Q = CV (Capacitors In Series And In Parallel) and energy formulas (Energy Stored In Capacitors)
  • Outcomes: find the final shared voltage/charges after connecting capacitors in parallel, and explain the “missing energy”
  • Key result (isolated parallel sharing): V_f = (C₁V₁ + C₂V₂)/(C₁ + C₂), Q_(i,f) = CᵢV_f (If C₂ starts uncharged, then V₂ = 0.)
  • Common trap: assuming electrostatic energy is conserved during redistribution (it typically decreases due to dissipation in the connection)

Example in this lesson:

  • C₁ = 8.0 μF initially charged to V₀ = 120 V, then disconnected from the battery.
  • Uncharged C₂ = 4.0 μF is connected in parallel to C₁.
  • Results: Q₀ = 960 μC, V_f = 80 V, Q₁ = 640 μC, Q₂ = 320 μC.

Motivation / intuition

When you connect capacitors, charges flow until both devices share the same final voltage (parallel constraint). The redistribution is a transient current process in real wires, so not all initial electrostatic energy remains stored.

2) Setup

Assumptions and sign conventions:

  • Capacitors are ideal.
  • Battery is removed before sharing, so total charge on connected capacitor system is conserved.
  • Both capacitors end with the same voltage in parallel.
  • Take charge magnitude as positive on one plate pair; opposite plate has equal negative charge.

Useful formulas:

Q = CV, U = (1/2)CV² = Q²/2C = (1/2)QV
Quick checks (units + limits/sign)
  • Units: Q = CV gives C = (F)(V) and U = 1/2 CV² gives J = (F)(V²).
  • Limits/signs: for isolated sharing, V_f must lie between the initial voltages; if C₂ → 0, then V_f → V₁; charge conservation means Qₜₒₜₐₗ before and after is the same.

3) Core derivation/explanation

Initial state:

Q₀ = C₁V₀ = (8.0 μF)(120 V) = 960 μC
U₀ = (1/2)C₁V₀² = (1/2)(8.0 × 10⁻⁶)(120)² = 0.0576 J

After connecting C₂ in parallel:

C_eq = C₁ + C₂ = 12.0 μF

Charge conservation gives

Qₜₒₜₐₗ = Q₀ = C_eqV_f ⇒ V_f = Q₀/(C₁ + C₂) = 80 V

Then

Q₁ = C₁V_f = (8.0 μF)(80 V) = 640 μC
Q₂ = C₂V_f = (4.0 μF)(80 V) = 320 μC

Check: Q₁ + Q₂ = 960 μC = Q₀.

Final energy:

U_f = (1/2)(C₁ + C₂)V_f² = (1/2)(12.0 × 10⁻⁶)(80)² = 0.0384 J

Energy decrease:

Δ U = U_f-U₀ = -0.0192 J

This missing energy is dissipated during transient current flow (mainly heat and some EM radiation).

Where did the energy go?

The “missing” energy is not destroyed: during the brief current pulse, it is dissipated in the connection resistance as heat and emitted as electromagnetic radiation. Ideal capacitors plus an ideal wire with zero resistance is an unphysical limit.

4) Worked example(s)

The full worked calculation above is the standard template for capacitor-sharing problems:

  1. Find initial Q₀ and U₀.
  2. Use C_eq and charge conservation for V_f.
  3. Find each final charge using Qᵢ = CᵢV_f.
  4. Compare initial and final energy.

5) Practice set (with hints + answers)

  1. If C₂ = 8.0 μF instead, what is V_f? Hint: V_f = Q₀/(C₁ + C₂) with same Q₀. Answer: V_f = 960 μC/16 μF = 60 V.

  2. In this lesson, is total charge conserved after disconnection from battery? Hint: isolated connected capacitor system. Answer: yes, total charge is conserved.

  3. Why is total electrostatic energy smaller after sharing? Hint: transient current in real connection path. Answer: some stored electrical energy is dissipated (mainly as heat).

6) Summary + next steps

  • Use Q = CV carefully; this is the most common algebra mistake.
  • In isolated parallel sharing, total charge is conserved and final voltage is common.
  • Energy usually decreases during redistribution, even when charge is conserved.

Next: Energy Stored In Spherical Capacitor Previous: Capacitance Of A Cylindrical Capacitor Back To Electromagnetism (UY1)

Continue with the next resource in this course.

Course and syllabus information
Course
University Physics Year 1
Edition
University Physics Year 1