Spherical Capacitor Calculator
Calculate capacitance of concentric spherical conducting shells
Calculate Spherical Capacitor
Radius of the inner conducting sphere
Radius of the outer conducting sphere
Material between the spherical shells
Dimensionless dielectric constant
Capacitance Results
Capacitor Analysis
Example Calculation
Van de Graaff Generator Sphere
Application: Electrostatic generator sphere
Inner radius: 10 cm
Outer radius: 15 cm
Dielectric: Air (εᵣ = 1.0006)
Calculation
C = 4πε₀εᵣ / (1/a - 1/b)
C = 4π × 8.854 × 10⁻¹² × 1.0006 / (1/0.1 - 1/0.15)
C = 1.112 × 10⁻¹⁰ / (10 - 6.667)
C = 1.112 × 10⁻¹⁰ / 3.333
C ≈ 33.4 pF
Typical capacitance for electrostatic demonstration equipment
Spherical Capacitor Formula
C = 4πε₀εᵣ / (1/a - 1/b)
Spherical capacitor equation
Common Dielectric Materials
Typical Capacitance Values
Electrostatic Generators
10-100 pF
High Voltage Capacitors
0.1-10 nF
Power Capacitors
1-100 μF
Energy Storage
0.1-10 mF
Capacitor Physics
Spherical symmetry provides uniform field
Higher εᵣ increases capacitance linearly
Smaller gap increases capacitance
Energy density highest near inner sphere
Breakdown voltage limits field strength
Understanding Spherical Capacitors
What is a Spherical Capacitor?
A spherical capacitor consists of two concentric conducting spheres separated by a dielectric material. The inner sphere carries a positive charge while the outer sphere carries an equal negative charge, creating a radial electric field between them.
Key Properties
- •Radially symmetric electric field
- •Field strength inversely proportional to r²
- •Maximum field at inner sphere surface
- •Energy stored in electric field
Mathematical Derivation
C = 4πε₀εᵣ/(1/a - 1/b)
From Gauss's law and potential difference
Physical Relations
Note: As b → ∞, the capacitance approaches 4πε₀εᵣa, representing an isolated sphere in an infinite medium.
Applications & Use Cases
Electrostatic Generators
Van de Graaff generators, Wimshurst machines, Tesla coils
High Voltage Systems
Power transmission, X-ray equipment, particle accelerators
Research Applications
Plasma physics, atmospheric research, electrostatic studies
Design Considerations
Maximizing Capacitance
- • Use high permittivity dielectric materials
- • Minimize gap between spheres (b - a)
- • Increase the size of both spheres proportionally
- • Consider temperature stability of dielectric
- • Account for frequency-dependent permittivity
Safety & Limitations
- • Dielectric breakdown limits maximum voltage
- • Corona discharge at sharp edges
- • Temperature effects on dielectric properties
- • Mechanical stress from electrostatic forces
- • Insulation requirements for high voltages