Capacitance and electrostatic energy
Define a two-conductor capacitance from the charge–potential-difference relation and derive plate, spherical and concentric-sphere results under explicit geometry assumptions. Build U=∫V dQ into ½CV² and the vacuum field-energy integral, separate point-source self-energy from finite distributions, and state how linear dielectrics, fixed-Q versus fixed-V operation, nonlinear response and dispersion change the energy bookkeeping.
- Define C=Q/ΔV with a declared electrode polarity and show why linear electrostatics makes C depend on geometry/material rather than operating charge.
- Derive C≈ε₀A/d for large parallel plates with the stated negligible-fringing approximation.
- Derive the capacitance of an isolated conducting sphere relative to infinity and that of a concentric spherical capacitor.
- Derive capacitor energy by reversible quasistatic charging U=∫₀^Q V(q)dq and show equivalence of ½CV², Q²/(2C) and ½QΔV in a linear device.
- Connect finite localized source energy to ½∫ρV dV and the vacuum field-energy integral ε₀/2∫E²dV under stated boundary and self-energy assumptions.
- Check the plate capacitor's field energy against the circuit energy numerically and dimensionally.
- Describe how fixed-Q and fixed-V conditions lead to different energy changes when a linear dielectric is inserted.
- State that ½E·D is the stored-energy density only under suitable linear, lossless, nondispersive, quasistatic assumptions.
- Distinguish nonlinear reversible charging energy ∫E·dD from ½E·D and identify why THz dispersive materials require a broader treatment.
- Solve capacitor geometry and energy exercises including validity checks for fringing, endpoints and ideal point-source divergence.
Capacitance describes how much equal-and-opposite electrode charge is required to create a potential difference in a declared electrostatic geometry. Energy follows from the work needed to assemble that charge. The familiar half factors follow from LINEAR charging, and require care in dispersive or nonlinear media.
05.1Define a two-electrode capacitance with polarities fixed
For two conductors in a fixed, linear electrostatic geometry, assign conductor 1 charge +Q, conductor 2 charge −Q (Q>0), and define the potential difference ΔV=V₁−V₂>0. The capacitance is
The relation scales linearly with charge because Coulomb/Poisson electrostatics and the declared constitutive law are linear. C is a geometry/material parameter in this model, not a universal constant independent of environment. If other conductors or stray couplings matter, a capacitance matrix or a specified return electrode replaces the oversimplified two-terminal picture.
With V1−V2=ΔV>0 and upper charge +Q, the ideal gap field points DOWNWARD, |E|≈ΔV/d. For large plates A at fixed d and negligible fringe fields, C≈ε0A/d and U≈½C(ΔV)2≈½ε0E2Ad. The drawing represents plate ELECTRODES and an electric field, not a stretched circuit capacitor symbol.
05.2Parallel plates: derive rather than memorize C≈ε₀A/d
Take two large parallel metallic plates of facing area A, gap d, charges ±Q, vacuum gap and negligible edge/fringing effects (lateral dimensions much greater than d). Each isolated ideal sheet has field magnitude σ/(2ε₀), but between opposite sheets their fields add while outside they approximately cancel:
C increases with plate area and decreases with gap. The approximation is not exact for finite plates near their edges; fringing adds capacitance and makes the field nonuniform. For an ideal linear homogeneous dielectric fully filling the gap, replace ε₀ by a suitably declared static ε, not by a frequency-dependent complex permittivity without an energy model.
05.3Isolated conducting sphere and the return at infinity
For an isolated conducting sphere of radius R with total charge Q and no nearby sources, Vsurface=kQ/R relative to V(∞)=0. Defining a capacitance relative to the infinitely distant return gives
This is not a literal two-finite-electrode capacitor; the return boundary is infinity. If other bodies are close, the potential for a given Q changes and so does this effective self-capacitance.
05.4Two concentric spherical conductors
An inner conducting sphere of radius a carries +Q, a concentric outer conducting shell of inner radius b>a carries −Q, and the intervening region is vacuum. Spherical symmetry and Gauss's law give E=kQ/r² r̂ for a<r<b. Integrate the radial field from the outer conductor to the inner to obtain
The limit agrees with the isolated sphere. The stated field and capacitance rely on concentric conducting equipotentials and a homogeneous gap; a displaced inner sphere requires a different boundary-value solution.
05.5Derive the energy from reversible charging work
Imagine charging a capacitor quasi-statically from zero to Q while keeping geometry and material fixed. If its instantaneous charge is q and voltage V(q), transferring an additional dq requires reversible electrical work dU=V(q)dq. For a LINEAR capacitor V(q)=q/C:
The factor 1/2 comes from the voltage rising linearly during charging, not from a general rule that every electrical system has half the final voltage times final charge. Leakage, hysteresis, nonlinear response and active sources require a new work balance.
05.6Connect charge assembly to vacuum field energy
For a FINITE, localized static charge distribution in vacuum, take the potential V produced by the entire distribution with a suitable reference and a regularized finite source model. Assembly energy counts pairs only once, which introduces the factor 1/2:
The second equality follows by integration by parts when the boundary term ε₀∮V E·dA vanishes at infinity (or is included correctly for finite boundaries). The self-energy of an IDEAL mathematical point charge diverges; do not apply the finite energy conclusion blindly to delta-function sources.
05.7Parallel-plate energy: an independent numerical check
For A=1.00 cm²=1.00×10⁻⁴ m², d=1.00 mm=1.00×10⁻³ m and vacuum, C≈ε₀A/d≈8.85×10⁻¹³ F=0.885 pF. At ΔV=10.0 V, E≈10⁴ V/m and the gap volume is Ad=10⁻⁷ m³. Both computations give
The equality is a stringent units and model check, not two independent types of stored energy to add. The field formula includes the neglected fringe region only approximately because the plate calculation neglected fringing from the outset.
05.8Linear dielectric: fixed charge and fixed voltage are different setups
For the same parallel-plate geometry fully filled by a linear, lossless, isotropic STATIC dielectric with εr=ε/ε₀, neglect fringing and interfacial voids. Then C=εrC₀. Inserting a dielectric at fixed isolated-electrode charge Q reduces the stored field energy to UQ=Q²/(2εrC₀). At fixed imposed voltage ΔV, the stored field energy rises to UV=(εrC₀)(ΔV)²/2, but the battery exchanges charge and energy with the system.
Do not infer a mechanical force from only the capacitor's stored energy at fixed V without accounting for source work. Both scenarios require the equilibrium dielectric model; neither formula describes arbitrary THz dispersive polarization.
05.9The scope of u=½E·D and a nonlinear charging path
In a local, linear, lossless, nondispersive medium undergoing quasi-static reversible charging, D=εE and the stored electric energy density can be written u=½E·D. For a reversible nonlinear relation, use the work integral along the actual charging path instead:
A nonlinear/hysteretic medium may not have a path-independent stored-energy function. With high-frequency dispersion and loss, substituting a complex ε(ω) into ½εE² is not a valid generic energy calculation; material degrees of freedom, frequency response, dissipation and the measurement conditions must be included in a broader electromagnetic-energy treatment.
05.10Mastery exercises with worked solutions
- Derive C≈ε₀A/d from the fields of two oppositely charged large sheets; state the approximation controlling this derivation.
- Compute C and U for vacuum plates A=1.00 cm², d=1.00 mm, ΔV=10.0 V. Compare with the field integral.
- Derive C for concentric spherical conductors a=0.020 m, b=0.050 m; calculate its numerical value and recover the isolated-sphere limit.
- An isolated conducting sphere has R=0.100 m, Q=+1.00 nC. Find C, Vsurface and U.
- For a linear dielectric with εr=4, compare the energy with vacuum at fixed Q and at fixed V; explain the role of the voltage source.
- Why does ∫ε₀E²/2 over all space diverge for an ideal point charge, and why does this not contradict finite real capacitor energy?
Solutions and validity checks
1. Two large sheets ±Q/A give E≈Q/(ε₀A) in the gap and ΔV≈Ed; hence C≈ε₀A/d. Neglected fringing requires d small compared with the plates' lateral dimensions.
2. C≈8.85×10⁻¹³ F; U≈4.43×10⁻¹¹ J. E≈10⁴ V/m and Ad=10⁻⁷ m³ give (ε₀/2)E²Ad≈4.43×10⁻¹¹ J.
3. C=4πε₀ab/(b−a)≈3.71×10⁻¹² F=3.71 pF. At b→∞, C→4πε₀a.
4. C=4πε₀R≈1.11×10⁻¹¹ F=11.1 pF. V=Q/C≈89.9 V; U=Q²/(2C)≈4.49×10⁻⁸ J.
5. C→4C₀. At fixed Q, U→U₀/4. At fixed V, U→4U₀ but the source supplies/absorbs energy during the change.
6. E≈kq/r² close to a point charge, so the radial field-energy integral behaves as ∫r²E²dr∝∫dr/r² and diverges at r=0. Real conductors have finite spatial charge distributions; an ideal lumped-capacitor model excludes this point-source pathology.
EM-01 now connects source charges → vector field → potential → Gauss flux → conductor/interface conditions → capacitance and energy. EM-02 begins from magnetic force and magnetostatics without importing electrostatic equilibrium assumptions into time-varying fields.