physics 3-3

Capacitance and Geometry

Units of Capacitance

• Measured in Farads (F): The farad is the unit of capacitance, defined as the ability to store one coulomb of charge per one volt potential difference. It is fundamentally linked to how different materials and geometries affect the storage of electrical energy.

Capacitance as a Geometric Property

• Independence from Charge Amount or Position: Capacitance is a property that does not depend on the amount of electric charge stored or the precise positioning of the charge. This means that regardless of how much charge is added to the capacitor, the capacitance remains constant as long as the physical configuration stays the same.

• Increased Charge Does Not Alter Capacitance: Even though adding charge increases the voltage, it does not change the ratio of charge to voltage that defines capacitance.

Example of Capacitance

• Setup: Consider two conducting spheres, one positively charged (+q) and the other negatively charged (-q), separated by a distance (d). An electric field is established between them, directed from the positive to the negative sphere.

• Capacitance Determination: The capacitance of this setup is impacted by the distance between the spheres. A larger separation results in lower capacitance because the potential drop (voltage) is larger for any given amount of charge. This can be understood through an analogy to gravitational potential, where the potential drop is related to vertical height and horizontal distance.

Electric Field Considerations

• Electric Field (E): This is defined as the force experienced per unit charge and plays a critical role in determining capacitance. The relationship can be mathematically expressed as E = ΔV/d, where ΔV is the potential difference and d is the distance.

• Parallel Plate Capacitors: The electric field between two parallel plates can be calculated with the formula E = Q/(2Aε₀), where Q represents the charge, A is the area of the plates, and ε₀ (permittivity of free space) is approximately 8.85 x 10⁻¹² F/m. This electric field directly contributes to the energy storage capacity between the plates.

Derivation of Capacitance

• Formula: The capacitance can be expressed with the formula C = Q/ΔV, where Q is the total charge and ΔV is the potential difference across the capacitor. Reviewing the units, we find that all meters cancel out, confirming that capacitance is indeed expressed in farads (F).

• Contributing Factors: The area of the capacitor plates and the distance between them have significant influence on capacitance; increasing the plate area enhances capacitance, while increasing the distance decreases it.

Real-World Example

• To achieve a capacitance of 1 farad with a separation of 1 mm between the plates, one would require an impractically large plate area of approximately 100 million square meters, equivalent to about 20,000 football fields. This highlights the rarity of very high capacitance capacitors in practical use.

• Typical Capacitor Values: Common capacitance values encountered in everyday applications typically range from the order of picofarads (10⁻¹² farads) to microfarads (10⁻⁶ farads). These smaller capacitance values serve functions in various electronic circuits, such as filtering and timing.

Dielectric Materials

• Dielectric Breakdown: The maximum electric field that air can support without discharging is approximately 3 million newtons per coulomb, which can lead to unwanted static electric discharges (static shocks).

• Dielectric Materials Introduction: When materials such as glass or water are introduced into the space between capacitor plates, they polarize under the influence of an electric field, which increases capacitance. These materials have specific dielectric constants (kappa) that quantify their effectiveness in this capacity.

Typical Kappa Values for Dielectrics:

• Glass: Kappa values range from 5 to 10.

• Water: Has a much higher kappa value of about 80, indicating its strong capacity to enhance capacitance when used as a dielectric.

Conclusion on Dielectrics

• By inserting a dielectric substance into a capacitor, the internal electric field strength is reduced, and the overall capacitance is effectively increased. This practical application is often utilized to enhance the performance of standard capacitors, allowing them to achieve desired capacitance values more efficiently by incorporating materials such as glass or other efficient dielectric substances in their design.