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Spherical Capacitor Formula Calculator

\[ C = 4 \pi \varepsilon_0 \frac{r_1 r_2}{r_2 - r_1} \]

F/m

1. What is the Spherical Capacitor Formula Calculator?

Definition: This calculator computes the capacitance (\(C\)) of a spherical capacitor, defined as \(C = 4 \pi \varepsilon_0 \frac{r_1 r_2}{r_2 - r_1}\), where \(\varepsilon_0\) is the permittivity of free space, and \(r_1\) and \(r_2\) are the radii of the inner and outer spheres, respectively.

Purpose: It is used in electromagnetism to determine the capacitance of a spherical capacitor, applicable in physics, electrical engineering, and capacitor design.

2. How Does the Calculator Work?

The calculator uses the spherical capacitor formula:

Formula: \[ C = 4 \pi \varepsilon_0 \frac{r_1 r_2}{r_2 - r_1} \] where:

  • \(C\): Capacitance (F, pF, nF, µF)
  • \(\varepsilon_0\): Permittivity of free space (F/m)
  • \(r_1\): Inner radius (cm, m, in, ft, yd)
  • \(r_2\): Outer radius (cm, m, in, ft, yd)

Unit Conversions:

  • Radius:
    • 1 cm = 0.01 m
    • 1 m = 1 m
    • 1 in = 0.0254 m
    • 1 ft = 0.3048 m
    • 1 yd = 0.9144 m
  • Capacitance (Output):
    • 1 F = 1 F
    • 1 pF = \( 10^{-12} \) F
    • 1 nF = \( 10^{-9} \) F
    • 1 µF = \( 10^{-6} \) F
The capacitance is calculated in farads (F) and can be converted to the selected output unit (F, pF, nF, µF). Results greater than 10,000 or less than 0.001 are displayed in scientific notation; otherwise, they are shown with 4 decimal places.

Steps:

  • Enter the permittivity of free space (\(\varepsilon_0\)), inner radius (\(r_1\)), and outer radius (\(r_2\)) with their units (default: \(\varepsilon_0 = 8.8541878128 \times 10^{-12} \, \text{F/m}\), \(r_1 = 0.1 \, \text{m}\), \(r_2 = 0.2 \, \text{m}\)).
  • Convert inputs to SI units (F/m, m), with independent unit selection for \(r_1\) and \(r_2\).
  • Validate that permittivity, inner radius, and outer radius are greater than 0, and the outer radius is greater than the inner radius.
  • Calculate the capacitance in farads using the formula.
  • Convert the capacitance to the selected output unit.
  • Display the result, using scientific notation if the value is greater than 10,000 or less than 0.001, otherwise rounded to 4 decimal places.

3. Importance of Spherical Capacitor Capacitance Calculation

Calculating the capacitance of a spherical capacitor is crucial for:

  • Physics: Understanding the storage of electrical energy in spherical geometries, often used in theoretical models and experiments.
  • Electrical Engineering: Designing capacitors with spherical configurations for specific applications, such as in high-voltage systems or sensors.
  • Education: Teaching the principles of capacitance and the effect of geometry on electric fields in spherical systems.

4. Using the Calculator

Examples:

  • Example 1: Calculate the capacitance for \(\varepsilon_0 = 8.8541878128 \times 10^{-12} \, \text{F/m}\), \(r_1 = 0.1 \, \text{m}\), \(r_2 = 0.2 \, \text{m}\), output in pF:
    • Enter \(\varepsilon_0 = 8.8541878128 \times 10^{-12} \, \text{F/m}\), \(r_1 = 0.1 \, \text{m}\), \(r_2 = 0.2 \, \text{m}\).
    • Convert: \(r_1 = 0.1 \, \text{m}\), \(r_2 = 0.2 \, \text{m}\).
    • Compute: Numerator: \(4 \pi \varepsilon_0 \times r_1 r_2 = 4 \pi \times 8.8541878128 \times 10^{-12} \times (0.1 \times 0.2) \approx 2.22583788576 \times 10^{-12}\).
    • Denominator: \(r_2 - r_1 = 0.2 - 0.1 = 0.1\).
    • Capacitance in F: \(C = \frac{2.22583788576 \times 10^{-12}}{0.1} \approx 2.22583788576 \times 10^{-11} \, \text{F}\).
    • Convert to output unit (pF): \(2.22583788576 \times 10^{-11} \times 10^{12} \approx 22.2583788576 \, \text{pF}\).
    • Result: \( \text{Capacitance} = 22.2584 \, \text{pF} \).
  • Example 2: Calculate the capacitance for \(\varepsilon_0 = 8.8541878128 \times 10^{-12} \, \text{F/m}\), \(r_1 = 10 \, \text{cm}\), \(r_2 = 4.724409449 \, \text{in}\), output in nF:
    • Enter \(\varepsilon_0 = 8.8541878128 \times 10^{-12} \, \text{F/m}\), \(r_1 = 10 \, \text{cm}\), \(r_2 = 4.724409449 \, \text{in}\).
    • Convert: \(r_1 = 10 \times 0.01 = 0.1 \, \text{m}\), \(r_2 = 4.724409449 \times 0.0254 = 0.12 \, \text{m}\).
    • Compute: Numerator: \(4 \pi \varepsilon_0 \times r_1 r_2 = 4 \pi \times 8.8541878128 \times 10^{-12} \times (0.1 \times 0.12) \approx 1.335502731456 \times 10^{-12}\).
    • Denominator: \(r_2 - r_1 = 0.12 - 0.1 = 0.02\).
    • Capacitance in F: \(C = \frac{1.335502731456 \times 10^{-12}}{0.02} \approx 6.67751365728 \times 10^{-11} \, \text{F}\).
    • Convert to output unit (nF): \(6.67751365728 \times 10^{-11} \times 10^9 \approx 0.0667751365728 \, \text{nF}\).
    • Result: \( \text{Capacitance} = 0.0668 \, \text{nF} \).

5. Frequently Asked Questions (FAQ)

Q: What is a spherical capacitor?
A: A spherical capacitor consists of two concentric spherical conductors separated by a dielectric (in this case, free space), with the capacitance determined by the radii of the spheres and the permittivity of the medium between them.

Q: Why must the outer radius be greater than the inner radius?
A: The outer sphere must enclose the inner sphere for the capacitor to function, requiring \(r_2 > r_1\); otherwise, the denominator becomes zero or negative, making the capacitance undefined or negative, which is physically meaningless.

Q: What is the permittivity of free space?
A: The permittivity of free space (\(\varepsilon_0\)) is a physical constant that relates the electric field to the charge in a vacuum, approximately \( 8.8541878128 \times 10^{-12} \, \text{F/m} \).

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