Energy stored in a capacitor

How does a capacitor store energy?

In plain words: the first bit of charge is easy to move, because the capacitor is still uncharged. Each later bit has to be pushed against the charge already there, so it costs more. The voltage rises steadily from 0 to V as the charge goes in, so on average each coulomb costs V/2, and the total is ½QV.

Energy as work done in charging

These formulas assume fixed geometry and a linear dielectric, so capacitance is constant. The integral gives reversible charging work stored in the capacitor; a real charging source may supply additional energy dissipated in the circuit.

• Start with two uncharged conductors and move positive charge from conductor 2 to conductor 1 bit by bit until conductor 1 has Q; by conservation of charge conductor 2 then has −Q. External work is needed at every stage, since conductor 1 is always at the higher potential (NCERT pages 73–74).
• At an intermediate stage, with charges Q′ and −Q′, the potential difference is V′ = Q′/C. Moving a further small charge δQ′ takes work (NCERT Eq. (2.68), page 74):

• Adding up these steps from 0 to Q (integrating) gives W = Q²/2C. Using Q = CV, the result can be written three ways (NCERT Eq. (2.69), page 74):

• Because the electrostatic force is conservative, the stored energy does not depend on how the charge was built up (page 74).
• On a graph of V′ against Q′ the line is straight, so the work is the area of the triangle under it: ½ × Q × V (reasoned from NCERT Eq. (2.68)).

Energy density of the electric field

NCERT shows that the energy can be thought of as stored in the field between the plates, using a parallel plate capacitor with plates of area A, a distance d apart (NCERT page 74):

• The energy is ½Q²/C. With Q = Aσ and C = ε₀A/d, this is (Aσ)²/2 × d/(ε₀A) (NCERT Eq. (2.70), page 74).
• With E = σ/ε₀ (NCERT Eq. (2.71)), the energy becomes U = ½ε₀E² × Ad (NCERT Eq. (2.72), page 74).
• Ad is the volume between the plates, the field-filled region in the ideal model that neglects fringing. So the energy per unit volume, the energy density, is (NCERT Eq. (2.73), page 75):

• NCERT derived this for a parallel plate capacitor, but it is very general: in vacuum it holds for the electrostatic field of any arrangement of charges (page 75).
• With a linear isotropic dielectric of constant K filling the gap, C = ε₀KA/d and σ = ε₀KE, and the same steps give u = ½ε₀KE² (reasoned from NCERT Eqs. (2.49)–(2.51) and (2.69)). For vacuum, K = 1.

Joining a charged capacitor to an uncharged one

NCERT Example 2.10 shows that energy is lost when charge is shared, even though no charge is lost (pages 75–76):

• A capacitor charged to V is disconnected and joined to an identical uncharged capacitor. Charge is conserved, so each ends with Q/2 at a common potential difference V/2.
• The total energy is then 2 × ½ × (Q/2) × (V/2) = QV/4, half the starting ½QV.
• The other half is lost during the brief transient current that flows from the first capacitor to the second, as heat and electromagnetic radiation (page 76).
• In general (reasoned from charge conservation and NCERT Eq. (2.69)): join C₁ at V₁ to C₂ at V₂, positive plate to positive plate. The charge is conserved, so the common potential is V = (C₁V₁ + C₂V₂)/(C₁ + C₂).
• The loss is ½C₁V₁² + ½C₂V₂² − ½(C₁ + C₂)V². Putting in V, this simplifies to C₁C₂(V₁ − V₂)² divided by 2(C₁ + C₂), which is never negative.
• If a positive plate is joined to a negative one, use C₁V₁ − C₂V₂ for the charge and (V₁ + V₂)² in the loss.

Solved examples

Common mistakes to avoid

Quick revision summary

• Charging takes work because each bit of charge is moved against the charge already there: δW = V′δQ′ = (Q′/C)δQ′.
• The stored energy is U = Q^²/(2C) = ½CV^² = ½QV, the triangle under the V′–Q′ line; it does not depend on how the charge was built up.
• The energy can be seen as stored in the field, with energy density u = ½ε_₀E^² in vacuum for any field; in a dielectric it is ½ε_₀KE^².
• Joining a charged capacitor to an identical uncharged one keeps the charge but halves the energy; the rest goes as heat and electromagnetic radiation.
• In general the energy lost on joining is C_₁C_₂(V_₁ − V_₂)^²/[2(C_₁ + C_₂)].

Questions students ask

Is the energy of a charged capacitor stored in its charges or in its electric field?

The energy of a charged capacitor can be described either way, because in electrostatics the two give the same total: NCERT finds it as the potential energy of the separated charges, Q^²/2C, and then shows that this equals ½ε₀E^² for each unit volume of the field between the plates (NCERT pages 74–75). No measurement on static charges can tell the two pictures apart. The field picture is the one that keeps working when fields change: electromagnetic waves carry energy from one place to another through empty space, as light carries it from the Sun to the Earth (reasoned from NCERT Chapter 8, page 207).

What are the SI unit and dimensional formula of energy density, and why is it the same as pressure?

The energy density of an electric field, u = ½ε₀E^², is measured in J m^⁻³ and has the dimensional formula [M L^⁻¹ T^⁻²], energy [M L^² T^⁻²] per volume [L^³] (NCERT page 75; see dimensional formulae). Since 1 J m^⁻³ = 1 N m^⁻², this is also the unit of pressure, and the match has a meaning: beyond the NCERT text, the field pulls outward on each square metre of a charged conductor’s surface with a force σ^²/2ε₀, exactly ½ε₀E^² for the field E = σ/ε₀ just outside (reasoned from NCERT page 63). This pull draws a capacitor’s plates together.

Why does the energy of an isolated charged capacitor increase when its plates are pulled apart?

The energy of an isolated charged capacitor rises when its plates are pulled apart because work must be done against their attraction, and that work is stored: Q stays fixed, so U = Q^²/2C grows as C = ε₀A/d falls (reasoned from NCERT pages 69 and 74). The field σ/ε₀ between the plates, and so the energy density, does not change; the extra energy is the same field filling a larger volume. For plates of area 0.025 m^² carrying ±1.0 × 10^⁻⁷ C, pulling them 2.0 mm further apart takes about 4.5 × 10^⁻⁵ J. With a battery connected, V stays fixed instead and U = ½CV^² falls.

Is energy stored in a capacitor deleted from NCERT or the JEE and NEET syllabus?

No, energy stored in a capacitor has not been deleted: it is NCERT Class 12 Physics Part I, Section 2.15 (pages 73–76 of the 2026–27 reprint), with NCERT Example 2.10 on sharing charge between two capacitors. The JEE Main syllabus ends Unit 11, Electrostatics, with “energy stored in a capacitor”, and the NEET (UG) 2026 syllabus ends its Unit 11 with “Energy stored in a capacitor.” The energy density ½ε₀E^² is part of the same NCERT section (page 75), though neither syllabus names it.

What is the difference between the energy stored in a capacitor and in an inductor?

A capacitor stores energy in the electric field between its plates, ½CV^², while an inductor stores energy in the magnetic field of its current, ½LI^² (NCERT Chapter 6, Eq. (6.17), page 169). NCERT compares the two in its Example 6.9: per unit volume the energy is ½ε₀E^² in an electric field and B^²/2μ₀ in a magnetic field, both proportional to the square of the field and valid in vacuum wherever such a field exists (NCERT Chapter 6, pages 169–170). An isolated charged capacitor keeps its energy as long as its charge stays, whereas an inductor holds energy only while the current flows.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. A 4.0 µF capacitor holds 200 µC. Find the energy stored.

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   U = Q^²/(2C) = (200 × 10⁻⁶)²/(2 × 4.0 × 10⁻⁶) = 5.0 × 10⁻³ J (NCERT Eq. (2.69), page 74).

2. Q2. By what factor does the energy change if the potential difference is doubled while the capacitance stays fixed?

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   It becomes four times as large, since U = ½CV^² (NCERT Eq. (2.69), page 74).

3. Q3. Where, according to NCERT, can the energy of a charged capacitor be thought to reside?

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   In the electric field between the plates, with energy density ½ε₀E² (NCERT pages 74–75).