Capacitance

(a) Simple Parallel Plate Capacitor

A parallel plate capacitor consists of two equal metal plates placed parallel to each other.

• The plates are separated by a vacuum or air (which acts as the dielectric medium in this case).

• The capacitance of such a capacitor depends on the surface area of the plates, the distance between them, and the properties of the dielectric medium (vacuum or air in this case).

(b) Capacitor Storing Energy

A capacitor stores energy by transferring charge from one plate to the other.

• The plates carry equal but opposite charges: one plate becomes positively charged, while the other becomes negatively charged.

• The net charge on the capacitor is zero, but there is an electric field between the plates due to the separated charges.

(c) Capacitance Definition

Capacitance (C) is defined as the ratio of the charge (Q) on one plate to the voltage (V) across the plates:

C = Q / V

Unit: farad (F),

where: 1 F = 1 coulomb per volt (C/V).

(d) Capacitance for a Parallel Plate Capacitor (No Dielectric)

For a parallel plate capacitor without a dielectric (i.e., a vacuum or air between the plates), the capacitance is given by:

C = ϵ₀ A / d

where:

• C = capacitance

• ϵ₀​ = permittivity of free space (8.85×10−12 F/m8.85×10−12F/m)

• A = area of one plate

• d = distance between the plates

(e) Effect of Dielectric on Capacitance

A dielectric material placed between the plates of a capacitor increases its capacitance.

The capacitance with a dielectric is given by:

C = κ ⋅ ϵ₀A / d

where κ is the relative permittivity (dielectric constant) of the material, which is greater than 1.

The dielectric reduces the electric field within the capacitor, allowing it to store more charge for the same applied voltage.

(f) Electric Field in a Parallel Plate Capacitor

The electric field (E) between the plates of a parallel plate capacitor is uniform (except at the edges of the plates) and is given by:

E = V / d

where:

• E = electric field

• V = potential difference across the plates

• d = distance between the plates

(g) Energy Stored in a Capacitor

The energy U stored in a capacitor is given by:

U= ½ QV

where:

• U = energy stored in the capacitor

• Q = charge on the capacitor

• V = voltage across the capacitor

(h) Capacitors in Series and Parallel

Capacitors in series: The total capacitance C_total of capacitors in series is given by:

1 / C_total = 1 / C₁ + 1 / C₂ +_⋯

Capacitors in parallel: The total capacitance C_total of capacitors in parallel is the sum of individual capacitances:

C_total = C₁ + C₂ +_⋯

(i) Capacitor Charging and Discharging Through a Resistor

A capacitor charges and discharges through a resistor in an RC circuit.

During charging, the voltage across the capacitor increases, while the current decreases over time.

During discharging, the voltage and charge on the capacitor decrease exponentially as the current flows through the resistor.

(j) Equations for Charging and Discharging Capacitors

Charging equation: The charge on a capacitor while charging through a resistor is given by:

Q(t) = Q₀ (1 − e ^{− t / RC})

where:

• Q₀ = maximum charge the capacitor can hold

• t = time

• R = resistance

• C = capacitance

• RC = time constant

Discharging equation: The charge on a capacitor while discharging through a resistor is given by:

Q(t) = Q₀e ^{− t / RC}

where the variables are the same as in the charging equation.