Cylindrical Capacitors and Coaxial Cables

Coaxial cables consist of a central conductor surrounded by a cylindrical shield separated by an insulating dielectric. This geometry forms a cylindrical capacitor that stores electric charge when a potential difference exists between the inner and outer conductors. The capacitance per length influences signal transmission, impedance, and energy storage properties of the cable. For a coaxial geometry with inner radius $a$, outer radius $b$, length $L$, and dielectric permittivity $\epsilon =\mathrm{\epsilon ₀}\mathrm{\epsilon ₛ}$, the capacitance is $C=\frac{\mathrm{2\pi \epsilon L}}{\text{ln}}$. This expression emerges from Gauss’s law applied to cylindrical symmetry and reveals how geometry and material determine capacitance.

Deriving the Capacitance Formula

Consider an inner cylinder carrying positive charge and an outer cylinder carrying equal negative charge. By symmetry, the electric field points radially outward and depends only on distance from the axis. Applying Gauss’s law to a cylindrical Gaussian surface of radius $r$ and length $L$ yields $E\left(r\right)=\frac{\lambda }{\mathrm{2\pi \epsilon r}}$, where $\lambda$ is charge per length. The potential difference between the conductors is the integral of the electric field from $a$ to $b$, giving $V=\frac{\lambda }{\mathrm{2\pi \epsilon }}\text{ln}\left(\frac{b}{a}\right)$. Capacitance, defined as $C=\frac{Q}{V}$, becomes $C=\frac{\mathrm{\lambda \; L}}{/}=\frac{\mathrm{2\pi \epsilon L}}{\text{ln}}$. This derivation underscores that capacitance scales linearly with length and permittivity but varies inversely with the logarithm of the radius ratio.

Exploring Variable Influence

The formula reveals several intuitive trends. Increasing the dielectric constant $\mathrm{\epsilon ₛ}$ raises capacitance by allowing more electric field lines to penetrate the insulating material. A shorter separation between conductors (smaller $ln\left(\frac{b}{a}\right)$) also increases capacitance, which is why cables designed for high capacitance per length use thin insulation. Extending the cable length $L$ proportionally increases total capacitance, an important consideration in long transmission lines where distributed capacitance affects signal timing and attenuation. Conversely, widening the gap between conductors or using a low-permittivity dielectric reduces capacitance, improving high-frequency performance at the expense of energy storage.

Worked Examples

Suppose a coaxial cable has inner radius 0.5 mm, outer radius 3 mm, length 1 m, and is filled with polyethylene with relative permittivity 2.3. Using the formula with $\mathrm{\epsilon ₀}=\mathrm{8.854\times 10⁻¹²}$ F/m, the capacitance is $C=\frac{\mathrm{2\pi \; \epsilon ₀\; \epsilon ₛ\; L}}{\text{ln}}$ ≈ 98 pF. If the dielectric is replaced with air ($\mathrm{\epsilon ₛ}=1$), the capacitance drops to about 43 pF. Doubling the length doubles the capacitance, while increasing the outer radius reduces it. These variations highlight how designers tailor cable dimensions and materials to achieve desired electrical characteristics.

Table of Relative Permittivities

Material	Relative Permittivity εᵣ	Typical Use
Vacuum	1.000	Reference
Air	1.0006	Radio frequency coax
Polyethylene	2.3	Common coax dielectric
PTFE (Teflon)	2.1	High-performance cables
Rubber	7	High-capacitance applications

Applications

Capacitance influences how coaxial cables transmit signals. In analog video or audio lines, high capacitance can attenuate high frequencies, leading designers to select materials and geometries that balance capacitance against durability and cost. In digital communications, distributed capacitance together with inductance determines characteristic impedance and signal velocity. Understanding the capacitance helps in calculating charge storage, energy transfer, and potential voltage rise during transient events. The same formula applies to any cylindrical capacitor, including laboratory instruments and certain types of sensors.

Assumptions and Limitations

This model assumes perfectly concentric cylinders, uniform dielectric, and negligible edge effects. Real cables may have braided shields, multiple dielectric layers, or slight eccentricities that alter capacitance. At very high frequencies, dielectric losses and skin effect complicate behavior. Nonetheless, the classical formula offers an excellent approximation for many engineering tasks and provides a foundation for more sophisticated analyses involving transmission line theory and electromagnetic wave propagation.

Using the Calculator

Enter values for any four of the five parameters—capacitance, length, inner radius, outer radius, or relative permittivity—and leave the remaining field blank. The script verifies that exactly one field is empty, then solves for it algebraically. When solving for a radius, it assumes the other radius is known and uses exponential functions derived from rearranging the formula. Ensure radii and length are in meters, capacitance in farads, and relative permittivity dimensionless. The tool safeguards against nonphysical inputs like negative lengths or a case where the outer radius is not larger than the inner radius.

Historical Background

The coaxial cable was invented in the late nineteenth century by Oliver Heaviside and later refined by engineers seeking to transmit radio frequency signals with minimal interference. Its cylindrical symmetry confines electric fields within the cable, reducing crosstalk and radiation. Over time, understanding the cable’s capacitance proved essential for impedance matching and signal integrity. The underlying mathematics draws from nineteenth-century electrostatics, demonstrating the enduring relevance of classical physics in modern technology. From early telegraph lines to contemporary broadband internet, the coaxial cable remains a workhorse of communication infrastructure.

Future Directions

As data rates increase, engineers explore novel dielectrics, air-spaced designs, and micro-coax structures on printed circuit boards. Understanding capacitance per unit length guides these innovations, enabling cables that support gigahertz frequencies with minimal loss. The same principles inform the development of superconducting transmission lines and quantum computing interconnects, where low capacitance and high impedance are crucial. By grasping the basic formula provided here, students and practitioners prepare to engage with cutting-edge advancements in electromagnetic design.