Spherical Capacitor
Calculate the capacitance of a concentric spherical capacitor using C = 4πε₀εᵣ(r₁r₂)/(r₂-r₁). Get capacitance, stored charge, and energy with interactive charts and breakdowns for physics and electronics.
About This Calculator
The Spherical Capacitor Calculator computes the capacitance, stored charge, and electrostatic energy of a concentric spherical capacitor. This calculator is ideal for physics students learning electromagnetism, electronics engineers designing high-voltage capacitors, and anyone studying the behavior of electric fields in spherical geometries.
A spherical capacitor consists of two concentric conducting spherical shells with a dielectric material filling the space between them. The capacitance depends on the radii of both spheres and the dielectric constant of the material. The formula used is C = 4\u03c0\u03b5\u2080\u03b5\u1d63 \u00d7 (r\u2081r\u2082)/(r\u2082 \u2212 r\u2081), where \u03b5\u2080 = 8.854 \u00d7 10\u207b\u00b9\u00b2 F/m is the vacuum permittivity, \u03b5\u1d63 is the relative permittivity, r\u2081 is the inner radius, and r\u2082 is the outer radius. When a voltage V is applied, the charge stored is Q = CV and the energy is E = \u00bdCV\u00b2.
The electric field between the spheres is radially symmetric and varies as E(r) = Q/(4\u03c0\u03b5\u2080\u03b5\u1d63r\u00b2). This non-uniform field distribution makes spherical capacitors useful in certain high-voltage applications where parallel-plate designs would suffer from edge effects. The breakdown voltage of a spherical capacitor depends on the dielectric strength of the material and the gap between the shells.
Regional Notes
India: Spherical capacitor concepts are covered extensively in CBSE Class 12 Physics and undergraduate engineering electromagnetics courses (B.Tech/B.E. programs).
United States: Taught in AP Physics C: Electricity and Magnetism, as well as introductory physics sequences at US colleges and universities.
United Kingdom: Covered in A-Level Physics and first-year undergraduate physics and electrical engineering courses across UK institutions.
Frequently Asked Questions
What is a spherical capacitor?
A spherical capacitor consists of two concentric spherical conducting shells separated by a dielectric material. The inner sphere carries charge +Q and the outer sphere carries charge -Q, creating an electric field confined to the region between them.
What is the formula for spherical capacitor capacitance?
The capacitance of a spherical capacitor is given by C = 4\u03c0\u03b5\u2080\u03b5\u1d63 \u00d7 (r\u2081r\u2082)/(r\u2082 \u2212 r\u2081), where \u03b5\u2080 is the vacuum permittivity (8.854 \u00d7 10\u207b\u00b9\u00b2 F/m), \u03b5\u1d63 is the relative permittivity of the dielectric, r\u2081 is the inner sphere radius, and r\u2082 is the outer sphere radius.
How does the dielectric material affect capacitance?
The dielectric material increases capacitance by a factor equal to its relative permittivity \u03b5\u1d63. For example, air or vacuum has \u03b5\u1d63 = 1, while common dielectrics like glass (\u03b5\u1d63 \u2248 5-10), mica (\u03b5\u1d63 \u2248 7), or ceramic (\u03b5\u1d63 \u2248 10-100) can significantly increase the capacitance.
What happens if the outer radius is not much larger than the inner radius?
When the outer radius is close to the inner radius (small gap), the spherical capacitor approximates a parallel-plate capacitor. The capacitance formula simplifies and the device behaves like two closely spaced plates. As the gap increases, the capacitance decreases.
How is energy stored in a spherical capacitor?
The energy stored in a spherical capacitor is given by E = \u00bdCV\u00b2, where C is the capacitance and V is the applied voltage. The energy is stored in the electric field between the concentric shells and depends on both the geometry and the dielectric material.
Can I calculate charge stored from capacitance and voltage?
Yes, the charge stored on a spherical capacitor is Q = CV, where C is the capacitance and V is the applied voltage. Our calculator computes both charge and energy automatically when you enter the voltage.
What units should I use for radius inputs?
You should enter the inner and outer radii in centimeters (cm). The calculator converts them to meters internally for the capacitance calculation and displays results in farads (F), picofarads (pF), or the most appropriate SI sub-unit.
How does the spherical capacitor differ from a parallel-plate capacitor?
Unlike parallel-plate capacitors which have uniform electric fields, spherical capacitors have a radially symmetric electric field that varies inversely with the square of the distance from the center. Spherical capacitors are useful in high-voltage applications where uniform field distribution is important.