Cylindrical Capacitor Calculator

The Cylindrical Capacitor Calculator computes capacitance of coaxial cylindrical conductors from inner and outer radii, length, and relative permittivity.

What Is a Cylindrical Capacitor Calculator?

A cylindrical capacitor consists of two coaxial conductors separated by an insulating medium. The inner conductor has radius a, and the outer conductor has inner radius b. The insulating medium has permittivity, which controls how much electric field it can support.

The calculator accepts geometry and material data. It returns capacitance, stored energy, electric field strength, charge, and related results. It also shows the intermediate steps so you can audit the derivation and confirm each assumption.

Engineers use this tool for quick cable sizing, insulation checks, and lab experiments. Students use it to connect Gauss’s law to practical design. In both cases, it reduces errors and speeds up iteration.

Cylindrical Capacitor Formulas & Derivations

The classic model is two long, concentric cylinders of length L with inner radius a and outer radius b. The region between the cylinders is filled with a homogeneous dielectric with permittivity ε = εr ε0, where εr is relative permittivity and ε0 is the vacuum permittivity. The key results follow from Gauss’s law and line integration of the electric field.

• Capacitance: C = 2π ε L / ln(b/a). This grows with ε and L, and falls with ln(b/a).
• Per-unit-length capacitance: C′ = 2π ε / ln(b/a). Useful for cables where L varies.
• Electric field between cylinders: E(r) = V / [r ln(b/a)] for a < r < b. Peak field occurs at r = a.
• Charge and voltage relation: Q = C V. Linear charge density λ = Q / L.
• Stored energy: U = (1/2) C V². Per unit length: U′ = (1/2) C′ V².
• From Gauss’s law: ∮ E · dA = λ/ε gives E(r) = λ / (2π ε r), which integrates to V = (λ / 2π ε) ln(b/a) and leads to C′ above.

These formulas assume long cylinders so end effects are negligible. They also assume perfect conductors and a lossless dielectric. The derivation uses only standard electrostatics and physical constants, which makes the model robust and quick to evaluate.

How the Cylindrical Capacitor Method Works

The method treats the system as radially symmetric. The electric field lines are radial, and equipotentials are coaxial cylinders. With this symmetry, Gauss’s law reduces to a one-dimensional relation in r, and the potential difference is a simple integral. The result is a logarithmic dependence on the radius ratio.

• Model the geometry with inner radius a, outer radius b, and an effective length L.
• Select the dielectric by specifying εr. The calculator multiplies εr by ε0 to get ε.
• Compute ln(b/a). This dimensionless term drives the capacitance and field strength.
• Use C = 2π ε L / ln(b/a) to get capacitance; then apply Q = C V for charge.
• Evaluate E(r) = V / [r ln(b/a)] as needed, checking the peak at r = a against breakdown.

Because the model is analytical, it runs instantly. It also highlights sensitivities: small changes in b/a or ε can strongly affect the outcome. That makes it ideal for parametric design sweeps.

What You Need to Use the Cylindrical Capacitor Calculator

Gather a few geometric and material inputs. These define the physical system and determine the calculation path. If you know voltage, you can get charge and energy. If you know charge, you can get voltage and energy.

• Inner conductor radius a (length unit, e.g., mm or m).
• Outer conductor inner radius b (length unit), with b > a.
• Effective length L (length unit). Use actual length if end effects are negligible.
• Relative permittivity εr of the dielectric (dimensionless). Typical plastics: 2 to 4.
• Either applied voltage V or total charge Q (pick one to define the operating point).
• Optional: breakdown strength of the dielectric to evaluate safety margins.

Ensure b is strictly greater than a. For accuracy, use L at least 10 times larger than the gap (b − a) to minimize fringing. If L is short or fringing matters, expect the result to understate capacitance slightly. Enter units consistently; the tool converts them internally to SI to avoid mistakes.

Using the Cylindrical Capacitor Calculator: A Walkthrough

Here’s a concise overview before we dive into the key points:

1. Open the tool and choose a solve mode: Capacitance, Voltage from Charge, or Charge from Voltage.
2. Enter a, b, and L with their units.
3. Enter εr, or pick a material from the library to autofill εr.
4. Enter either V or Q, depending on the solve mode.
5. Press the Calculator action to compute outputs.
6. Review the results panel for C, Q, V, U, and Emax at r = a.

These points provide quick orientation—use them alongside the full explanations in this page.

Example Scenarios

Coaxial signal cable sizing: Let a = 0.5 mm, b = 2.0 mm, L = 1.0 m, and polyethylene with εr = 2.25. The per-length capacitance is C′ = 2π ε / ln(b/a) ≈ 90 pF/m, so C ≈ 90 pF. With V = 10 V, Q = C V ≈ 900 pC and U = (1/2) C V² ≈ 4.5 nJ. What this means: This geometry yields low capacitance per meter, suitable for fast signal edges and minimal loading.

Compact high-voltage tube capacitor: Take a = 5 mm, b = 15 mm, L = 0.20 m, and transformer oil with εr = 2.2. Capacitance is C ≈ 22.3 pF. At V = 20 kV, Q ≈ 0.446 μC and U ≈ 4.46 mJ. Peak field at r = a is Emax = V / [a ln(b/a)] ≈ 3.64 MV/m, typically below oil breakdown. What this means: The design is feasible at 20 kV with a safe field margin, though manufacturing tolerances still need checking.

Assumptions, Caveats & Edge Cases

The analytical model is elegant and fast, but it rests on several assumptions. Know them to judge applicability and confidence in the numbers.

• Long-cylinder approximation: L must be large compared with the gap (b − a) to neglect fringing.
• Homogeneous dielectric: The tool treats ε as uniform. Multilayer dielectrics require a modified derivation.
• Frequency effects: εr can vary with frequency and temperature; losses introduce complex permittivity.
• Breakdown and partial discharge: Always compare Emax to dielectric strength with margin for defects and humidity.
• Numerical sensitivity: If b ≈ a, ln(b/a) becomes small and C grows; round-off can magnify input errors.

When any assumption fails, consider end corrections, guard rings, or numerical simulation. A quick finite element check is wise for short or segmented designs. Use the calculator as a baseline, then refine.

Units & Conversions

Capacitance depends on length scales and permittivity, so unit consistency is essential. The calculator converts everything to SI internally. Still, it helps to know the common conversions and how to read results expressed in F, V, and N·m per Coulomb.

Read the table left to right: identify the quantity, note common display units, then convert to SI. The calculator displays both base units and your chosen units so you can cross-check the result quickly.

Tips If Results Look Off

Surprising values often trace to a unit mismatch, an inverted radius order, or an unrealistic εr. Work through these checks before changing the design.

• Confirm b > a and both are positive real numbers.
• Verify units on a, b, L. Millimeters accidentally entered as meters cause huge errors.
• Check εr against a materials table; most plastics fall between 2 and 4.
• If L is short, expect the analytical C to be slightly low due to fringing.
• Recalculate C′ for a 1 m length to compare against cable datasheets.

If numbers still look wrong, try a different solve mode. For instance, set V and compute Q, then invert using Q/C to confirm. The derivation panel shows each constant and step so you can spot a mismatch.

FAQ about Cylindrical Capacitor Calculator

Can this handle multilayer dielectrics?

The base mode assumes a single uniform ε. For multilayers, model each annular layer and combine using series capacitance of coaxial shells, or use a dedicated multilayer solver.

How do I get per-unit-length capacitance for a cable datasheet?

Enter a and b with any L, then read C′ from the outputs. C′ is independent of L and is reported in F/m for easy comparison.

Does it account for fringing at the ends?

No. The long-cylinder approximation ignores fringing. If L is not much larger than b − a, apply an end correction or validate with simulation.

Are the results valid at high frequency?

Capacitance holds, but εr and loss tangent vary with frequency. For RF work, use frequency-dependent material data and consider skin effect and dispersion.

Key Terms in Cylindrical Capacitor

Capacitance

The ability to store electric charge per unit voltage, measured in farads. For coaxial cylinders, it depends on geometry and permittivity.

Permittivity

A material property that measures how an electric field interacts with the medium. Absolute permittivity ε equals εr times ε0.

Relative Permittivity

A dimensionless ratio comparing a material’s permittivity to vacuum permittivity. It scales capacitance directly in the formulas.

Linear Charge Density

Charge per unit length on the inner conductor, denoted λ. It relates to the field by Gauss’s law in cylindrical symmetry.

Gauss’s Law

A Maxwell equation stating that electric flux through a closed surface equals enclosed charge divided by permittivity. It provides E(r) in the derivation.

Fringing Field

The non-uniform field at the capacitor ends where lines of force bow outward. It increases capacitance slightly beyond the ideal model.

Breakdown Strength

The maximum electric field a dielectric can withstand without electrical failure. Designs must keep Emax below this value with margin.

Natural Logarithm

The logarithm base e, written ln. The capacitance depends on ln(b/a), which captures the radial scaling in cylinders.

References

Here’s a concise overview before we dive into the key points:

• HyperPhysics: Capacitance of a Cylindrical Capacitor
• Wikipedia: Coaxial Cable Capacitance
• MIT OpenCourseWare: 8.02 Electrostatics Notes
• NIST CODATA: Vacuum Permittivity ε0
• Engineering ToolBox: Relative Permittivity (Dielectric Constants)

These points provide quick orientation—use them alongside the full explanations in this page.