The capacitance of a spherical conductor is directly proportional to its radius, epsilon-naught (the permittivity of free space), and the permittivity of the dielectric material inside the sphere. Inversely, it is inversely proportional to the distance between the sphere and any surrounding conductors.
Capacitance of a Sphere
The capacitance of a sphere, denoted as C, represents its ability to store electrical charge. It is directly proportional to the radius of the sphere (r) and the permittivity of the surrounding medium (ε):
C = 4 * π * ε * r
The permittivity (ε) varies depending on the material surrounding the sphere. For example, the permittivity of vacuum is approximately 8.85 x 10^-12 farads per meter (F/m).
Factors Affecting Capacitance of a Sphere:
- Radius (r): The capacitance increases linearly with the radius of the sphere. As the radius increases, the surface area increases, providing more space for charge storage.
- Permittivity (ε): The capacitance is directly proportional to the permittivity of the surrounding medium. Materials with higher permittivity, such as water, enhance the capacitance of the sphere.
Example:
Consider a spherical capacitor with a radius of 10 cm (0.1 m) in vacuum (ε = 8.85 x 10^-12 F/m). Using the formula above, the capacitance can be calculated as:
C = 4 * π * 8.85 x 10^-12 * 0.1
C ≈ 1.11 x 10^-10 F
Table of Capacitance Values:
The following table provides the capacitance values for spherical capacitors with different radii in various media:
| Radius (cm) | Permittivity | Capacitance (F) |
|---|---|---|
| 10 | Vacuum | 1.11 x 10^-10 |
| 10 | Water | 8.94 x 10^-10 |
| 15 | Vacuum | 1.66 x 10^-10 |
| 15 | Oil | 4.47 x 10^-10 |
It’s important to note that the capacitance of a sphere is independent of the charge stored on it. However, if the charge is increased, the potential difference (voltage) across the sphere will also increase proportionally.
Question 1: What is the capacitance of a sphere?
Answer: The capacitance, C, of a sphere is the amount of electric charge, Q, that can be stored on its surface when its electric potential, V, is increased by one volt. This can be expressed by the formula: C = Q / V. The capacitance of a sphere in a vacuum is directly proportional to its radius, r, and is given by the formula: C = 4πε₀r, where ε₀ is the vacuum permittivity.
Question 2: How does the presence of a dielectric material affect the capacitance of a sphere?
Answer: The presence of a dielectric material between the plates of a capacitor increases its capacitance. This is because the dielectric material reduces the electric field between the plates, which allows more charge to be stored on the plates for a given voltage. The increase in capacitance is proportional to the dielectric constant, κ, of the material.
Question 3: What are some factors that affect the capacitance of a sphere?
Answer: The capacitance of a sphere is affected by several factors, including its radius, the presence of a dielectric material, and the distance between the sphere and any other conductors. The capacitance increases with increasing radius, increasing dielectric constant, and decreasing distance to other conductors.
That’s it for the capacitance of a sphere! I hope this article has helped you understand this fundamental concept. If you have any further questions, don’t hesitate to reach out.
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