Capacitor

Capacitor: Concepts and Formulas

A capacitor is a two-terminal passive electrical device that stores energy in the form of an electric field. It is widely used in electronic circuits for storing energy temporarily, filtering signals, and managing timing operations.

Charge and Voltage Relationship

The charge stored in a capacitor is directly proportional to the voltage across its plates:

q ∝ V → q = CV

Where:

• q = charge on the plates
• V = voltage difference across the two plates
• C = capacitance of the capacitor

Capacitance and Its Unit

Capacitance refers to the capacity of a capacitor to hold electric charge in relation to the applied voltage:

C = q / V

Its SI unit is the Farad (F), where:

1 F = 1 Coulomb/Volt

Since 1 Farad is very large, practical capacitors are usually measured in microfarads (μF), nanofarads (nF), or picofarads (pF).

Electric Field Between Plates

The electric field E between the plates of a parallel plate capacitor is given by:

E = V / d = σ / (ε₀ε_r) = q / (A ε₀ ε_r)

Where:

• V = potential difference
• d = separation between the plates
• σ = q / A = surface charge density
• ε₀ = vacuum permittivity
• ε_r = relative permittivity (dielectric constant)
• A = area of each plate

Deriving Capacitance from Field Equation

Using C = q / V and substituting from the electric field expression:

C = ε₀ε_r A / d

If the space between the plates is vacuum or air (ε_r = 1), then:

C = ε₀ A / d

General Case with Multiple Dielectric Layers

For capacitors with layered dielectrics of varying thickness, the effective capacitance is:

C = ε₀ A / (Z·d / ε_r)

Where Z·d / ε_r represents the effective dielectric-adjusted thickness.

Capacitance of Multi-Plate Capacitors

For capacitors with n plates connected alternately, the equivalent capacitance becomes:

C = (n − 1) ε₀ ε_r A / d

Capacitance of a Spherical Conductor

For an isolated spherical conductor of radius r in a medium with permittivity ε_r, the potential is:

V = q / (4π ε_r r)

Therefore, the capacitance is:

C = q / V = 4π ε_r r

Hence, capacitance is directly proportional to the radius:

C ∝ r

Summary of Key Formulas

• q = CV
• C = ε₀ A / d (for air)
• C = ε₀ ε_r A / d (with dielectric)
• C = (n − 1) ε₀ ε_r A / d (multi-plate capacitor)
• E = V / d = q / (A ε₀ ε_r)
• C = 4π ε_r r (spherical conductor)

Understanding these fundamental equations and their derivations is essential for mastering capacitor-related problems. Practice numerical applications of these formulas to build a strong conceptual foundation.