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UY1: Energy Stored In Spherical Capacitor

Key idea: Derive spherical-capacitor energy using both capacitance and field-energy-density methods, with consistent assumptions and checks.

  • 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 Energy Stored In Spherical Capacitor. 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

U = (Q²/8πε₀)(1/a-1/b)
  • Common trap: integrating energy density outside the gap (the field exists only for a < r < b in the ideal model)
  • You can derive the same expression either from U = Q²/(2C) or by integrating u = (1/2)ε₀E².

Motivation / intuition

This is a good “consistency check” example: circuit-level energy formulas (U = Q²/2C) and field-level energy density (u = (1/2)ε₀E²) must agree when you apply the same assumptions.

2) Setup

Assumptions:

  • Electrostatic equilibrium and ideal conducting shells.
  • Vacuum permittivity ε₀ in the gap.
  • Electric field exists only in a < r < b.
  • Outward radial direction is positive.

Known capacitance for spherical capacitor:

C = 4πε₀ab/(b-a)

(from Capacitance Of Spherical Capacitor).

3) Core derivation/explanation

Method A: using capacitance

U = Q²/2C = Q²/(2(4πε₀ab/(b-a))) = (Q²/8πε₀)(b-a)/ab
U = (Q²/8πε₀)(1/a-1/b)

Method B: integrating field energy density

For a < r < b:

E(r) = Q/4πε₀r²

Energy density:

u = (1/2)ε₀E² = Q²/32π²ε₀r⁴

Total energy in the gap:

U = ∫ₐ^b u dV = ∫ₐ^b u (4π r²dr)
U = Q²/8πε₀∫ₐ^bdr/r² = (Q²/8πε₀)(1/a-1/b)

Both methods agree.

Quick checks (units + limits/sign)
  • Units: [Q²/(8πε₀a)] = J, so U is energy.
  • Limits/signs: U ≥ 0 always; if b → ∞, then U → Q²/(8πε₀a) (isolated charged sphere); if Q → 0, then U → 0.

4) Worked example(s)

Let Q = 5.0 nC, a = 0.040 m, b = 0.060 m.

U = (Q²/8πε₀)(1/a-1/b)
= (((5.0 × 10⁻⁹)²)/8πε₀)(25.0-16.67) ≈ 9.36 × 10⁻⁷ J
U ≈ 0.936 μJ

5) Practice set (with hints + answers)

  1. If b → ∞ (isolated charged sphere), what does the energy expression become? Hint: drop 1/b term. Answer: U → Q²/8πε₀a.

  2. If the shell gap increases (larger b with fixed a,Q), does U increase or decrease? Hint: inspect 1/a-1/b. Answer: increases, approaching Q²/(8πε₀a).

  3. Why is there no electrostatic energy contribution inside the conducting material? Hint: field inside ideal conductor in electrostatics. Answer: E = 0 inside conductor, so u = (1/2)ε₀E² = 0 there.

6) Summary + next steps

  • Spherical-capacitor energy can be derived consistently by two methods.
  • Final expression:
U = (Q²/8πε₀)(1/a-1/b)
  • Always state geometry assumptions and region where field exists.

Next: Dielectrics In Capacitors Previous: Transferring Charge And Energy Between Capacitors 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