Calculate the capacitance of a spherical capacitor based on its inner and outer radii and the dielectric material.
A spherical capacitor consists of two concentric spherical conducting shells separated by a dielectric material. It is a fundamental model in electrostatics used to understand how geometry affects the storage of electric charge.
The capacitance depends on the radii of the two spheres and the permittivity of the material between them. As the gap between the spheres (b - a) decreases, the capacitance increases.
Where:
• C is the capacitance (F)
• ε₀ is the vacuum permittivity (≈ 8.854 × 10⁻¹² F/m)
• εᵣ is the relative permittivity (dielectric constant)
• a is the inner radius (m)
• b is the outer radius (m)
If the outer radius (b) is taken to be infinity, the capacitance of a single isolated sphere is C = 4πε₀εᵣa. For Earth (a ≈ 6371 km), this is about 710 µF.
Dielectric materials increase the capacitance by a factor of εᵣ. They also prevent the two conducting shells from touching and causing a short circuit.
ε₀ (epsilon-naught) is the vacuum permittivity, a physical constant that represents the capability of a vacuum to permit electric field lines.
The capacitance is inversely proportional to the gap (b - a). A smaller gap means a stronger electric field for the same voltage, allowing more charge to be stored.
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