Concentric-sphere cell capacitance
How much capacitance does a sphere inside a sphere hold, with this water between them?
- C
- Capacitance, F
- ε
- Relative permittivity
- a
- Inner diameter, in
- b
- Outer diameter, in
LaTeX
C = 4\pi \varepsilon_0 \cdot ε \cdot \frac{a \cdot b}{2 \cdot (b - a)}Method
- Take the water's static permittivity and adjust it for temperature, exactly as the coaxial cell calculation does.
- Convert both diameters to metres and halve them for the radii r_a and r_b.
- The capacitance of concentric spheres is 4πε₀εᵣ · r_a r_b ÷ (r_b − r_a). Written in diameters, the product over the difference picks up a factor of a half, which is where the 2 in the formula comes from.
Assumptions
- Two complete concentric spheres. A real cell has a stem, a filler port and probably a flat, none of which this counts.
- The water is a pure dielectric. For anything but distilled water it conducts, and the cell is a resistor in parallel with this capacitance.
- This geometry is not in Meyer's patents. It is here because people build it; the physics is the textbook result and the archive makes no claim for the shape.