EM fields lecture 21 - examples of capacitor and calculation of capacitance

Overview

Tim Gorringe explains capacitance as charge divided by potential difference, first connecting it to geometry-dependent coefficients of potential and then calculating it directly from electric fields. He derives capacitance for an isolated sphere, concentric spherical conductors, parallel plates, and concentric cylinders, highlighting limiting cases and why field-based integration is useful for idealized infinite geometries.

Key takeaways

An isolated conducting sphere of radius A has self-capacitance C = 4πϵ₀A, so doubling its radius doubles its charge storage at a fixed potential.
For a concentric spherical capacitor with inner radius A and shell inner radius B, C = 4πϵ₀AB/(B − A); the shell's outer radius does not matter because the electric field vanishes outside the shell.
When the spherical gap is small relative to the sphere radius, its capacitance reduces to the parallel-plate form C ≈ ϵ₀S/D.
An ideal parallel-plate capacitor with area A and separation D has C = ϵ₀A/D, derived by combining the fields of the two plates and integrating across the gap.
For concentric cylinders with radii A and B, capacitance per unit length is 2πϵ₀/ln(B/A), reflecting the cylindrical field's 1/ρ dependence.
Potential-coefficient methods capture self- and mutual interactions for finite conductors, while direct field integration is especially practical for idealized infinite plates and cylinders.

Chapters

0:00 Two Capacitor Categories and Geometry-Dependent Potential Coefficients

Capacitance can describe either one isolated conductor or a pair of conductors charged equally and oppositely.
Coefficients of potential encode conductor geometry and relate conductor charges to their equipotential values.
For a two-conductor capacitor, the relevant voltage is the potential difference between conductors; for an isolated conductor, it is measured relative to a distant reference or ground.

3:00 Self-Capacitance as the Inverse of P₁₁

For one isolated conductor, the potential and charge are related by the single coefficient P₁₁.
Self-capacitance is C = Q/φ = 1/P₁₁, so it depends on the conductor's geometry rather than the chosen charge.
Unlike a two-plate capacitor, this case involves only the conductor's self-interaction.

6:30 Conducting Sphere: C = 4πϵ₀A

A conducting sphere of radius A has coefficient of potential P₁₁ = 1/(4πϵ₀A).
Taking the inverse gives the sphere's self-capacitance, C = 4πϵ₀A.
A larger sphere stores more charge for a given potential because its charge can spread over a larger surface.

8:30 Two-Conductor Capacitance from the Pij Matrix

Two conductors require four potential coefficients: P₁₁ and P₂₂ for self-effects, and P₁₂ and P₂₁ for mutual effects.
Symmetry gives P₁₂ = P₂₁; with charges +Q and −Q, the potential difference depends on P₁₁ + P₂₂ − 2P₁₂.
The mutual capacitance is Q/Δφ, determined by the combination of all three distinct geometry-dependent coefficients.

15:50 Concentric Spherical Conductors: Coefficient-Based Result

The example uses an inner sphere of radius A and a concentric conducting shell with inner radius B and outer radius C.
The coefficients are P₁₁ = (1/4πϵ₀)(1/A + 1/C − 1/B), while P₁₂, P₂₁, and P₂₂ equal 1/(4πϵ₀C).
Substitution into the two-conductor formula gives C = 4πϵ₀AB/(B − A); the shell's outer radius C cancels because the field is confined between A and B.

23:28 Re-deriving Spherical Capacitance by Integrating the Field

With charges +Q and −Q, spherical symmetry gives a radial field of magnitude Q/(4πϵ₀r²) in the gap between radii A and B.
Integrating the field from the outer shell at B inward to the sphere at A yields Δφ = Q(B − A)/(4πϵ₀AB).
Dividing Q by this potential difference reproduces C = 4πϵ₀AB/(B − A), independently of the coefficient matrix.

30:50 Spherical Capacitor Limits: Parallel Plates and an Isolated Sphere

When the gap D = B − A is much smaller than A, the spherical result approaches C ≈ ϵ₀S/D, where S is the approximately common surface area.
This thin-gap limit matches a parallel-plate capacitor folded into a spherical shape.
When B is much larger than A, the result approaches 4πϵ₀A, the self-capacitance of an isolated sphere; Earth and the ionosphere are described as a thin-gap spherical capacitor.

36:33 Parallel-Plate Capacitor: C = ϵ₀A/D

Each oppositely charged plate produces a field of magnitude σ/(2ϵ₀); between the plates, the fields add to E = σ/ϵ₀.
Integrating across plate separation D gives a potential difference Δφ = σD/ϵ₀.
Using Q = σA for plate area A yields C = ϵ₀A/D; the result has farad units because ϵ₀ is in farads per meter and A/D is in meters.

44:45 Concentric Cylinders: Logarithmic Field and Capacitance per Length

For an inner cylinder of radius A and a concentric shell with inner radius B, the radial field scales as λ/(2πϵ₀ρ).
Integrating from B to A gives Δφ = [λ/(2πϵ₀)] ln(B/A).
Because an infinitely long arrangement has infinite total capacitance, the useful result is capacitance per unit length: C/L = 2πϵ₀/ln(B/A).

51:00 Comparing Coefficient and Field-Integration Methods

Potential coefficients provide a direct route when the geometry is finite and the coefficients are known.
Directly integrating the electric field gives the voltage between conductors and works naturally for ideal infinite sheets and cylinders.
The examples conclude with the spherical, parallel-plate, and cylindrical capacitance formulas, each determined by geometry rather than the amount of charge applied.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, tim gorringe.