Capacitance Calculator
Calculated Capacitance:
8.02 pF
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What is a Cylindrical Capacitor?
A cylindrical capacitor is a fundamental electrical component used to store energy in an electric field. It consists of two concentric conducting cylinders separated by a dielectric material. The inner cylinder has a radius a, and the outer cylinder has a radius b. Both cylinders have a length L.
Its ability to store charge, its capacitance (C), depends entirely on its geometry and the dielectric material between the cylinders. The relationship is described by the formula:
Explore the diagram to see how these components fit together. Hover over each label to highlight the corresponding part of the capacitor.
The Physics Behind It
The formula for capacitance isn't arbitrary; it's derived from fundamental principles of electromagnetism. Follow the steps below to see how we arrive at the final equation using Gauss's Law.
Step 1: Find the Electric Field (E) using Gauss's Law
First, we imagine a Gaussian surface (a cylinder with radius r where a < r < b) and apply Gauss's Law. This allows us to find the electric field E at any point between the conducting cylinders. The law relates the electric flux through a closed surface to the charge enclosed.
Step 2: Calculate Potential Difference (V)
Next, we find the potential difference (voltage) V between the inner and outer cylinders by integrating the electric field E along a path from radius a to b. The potential difference is the work done per unit charge to move a charge between the two cylinders.
Step 3: Use the Definition of Capacitance (C = Q/V)
Finally, we use the fundamental definition of capacitance: the ratio of the magnitude of the charge Q on one conductor to the potential difference V between them. By substituting our expression for V from Step 2, the charge Q cancels out, leaving us with the final formula that depends only on the capacitor's physical properties.
Real-World Applications
The principles of the cylindrical capacitor are not just academic. They are critical to the technology we use every day.
Coaxial Cables
The most common application is the coaxial cable, used for transmitting high-frequency signals for cable television, internet, and other data communications. The cable's central wire acts as the inner cylinder, and the woven mesh shield acts as the outer cylinder. The material separating them is the dielectric. The cable's capacitance per unit length is a crucial characteristic that determines its performance, affecting signal integrity and impedance matching. Understanding this capacitance is essential for designing communication systems that minimize signal loss and distortion.
Cylindrical Capacitance: Inputs, Assumptions, and Limits
For two centered coaxial cylinders, the ideal capacitance is C = 2πεL / ln(b/a), where a is the outer radius of the inner conductor, b is the inner radius of the outer conductor, L is their overlap, and ε = ε0εr. The geometry requires b > a.
- Units: use one length unit consistently; SI dimensions with permittivity in F/m produce capacitance in farads.
- Dielectric: enter the relative permittivity for the actual material and relevant frequency, not an assumed air value.
- Edge effects: the formula neglects fringing and is most accurate when the overlap is long compared with the radial gap.
- Boundaries: do not use the centered formula for eccentric cylinders. Off-center conductors need the appropriate eccentric solution; several inner conductors generally require a capacitance-matrix or numerical field calculation.
