Capacitors

Topics covered: Capacitance, parallel plate capacitor, spherical and cylindrical capacitors, grouping of capacitors, RC circuits, energy stored, dielectrics.

1. Capacitance

Capacitance is the ability of a conductor to store electric charge.

C = Q / V

SI unit: farad (F)

Factors Affecting Capacitance

• Presence of nearby conductors
• Surface area of the conductor
• Presence of dielectric medium

2. Parallel Plate Capacitor

Consists of two parallel plates separated by distance d.

Surface charge density:

σ = Q / A

Electric field between plates:

E = σ / ε₀

Capacitance:

C = ε₀A / d

With Dielectric Slab (thickness t)

C = ε₀A / (d − t + t / K)

If dielectric completely fills the gap (t = d):

C = Kε₀A / d

3. Spherical Capacitor

Two concentric spheres of radii a and b (a < b).

Capacitance:

C = 4π ε₀ (ab / (b − a))

With dielectric medium:

C = 4π ε₀K (ab / (b − a))

4. Cylindrical Capacitor

Two concentric cylinders of radii a and b, length l.

Capacitance:

C = (2π ε₀ l) / ln(b / a)

5. Grouping of Capacitors

Series Combination

• Same charge on each capacitor
• Total voltage is sum of individual voltages

1 / C_eq = 1 / C₁ + 1 / C₂ + 1 / C₃

Voltage distribution: V ∝ 1 / C

Energy distribution: U ∝ 1 / C

Parallel Combination

• Same potential difference across each capacitor
• Total charge equals sum of charges

C_eq = C₁ + C₂ + C₃

Charge distribution: Q ∝ C

Energy distribution: U ∝ C

6. RC Circuits

Charging of Capacitor

Charge at time t:

q(t) = Cε₀ (1 − e^−t/RC)

Current:

i(t) = (ε₀ / R) e^−t/RC

Voltage across capacitor:

V(t) = ε₀ (1 − e^−t/RC)

Time constant:

τ = RC

After one time constant, charge = 63% of maximum value.

Discharging of Capacitor

Charge:

q(t) = q₀ e^−t/RC

Voltage:

V(t) = ε₀ e^−t/RC

After one time constant, charge reduces to 37%.

7. Energy Stored in a Capacitor

U = ½ CV²

U = ½ QV

U = Q² / (2C)

8. Dielectrics

Dielectrics are insulating materials that transmit electric effects without conduction.

Types of Dielectrics

• Polar dielectrics: Have permanent dipole moment (e.g. H₂O, HCl)
• Non-polar dielectrics: Dipole induced only in electric field (e.g. N₂, O₂)

Polarisation of Dielectric

Induction of equal and opposite charges on dielectric faces under electric field.

Dielectric constant:

K = (Electric field in air) / (Electric field in medium)

Dielectric breakdown: At very high electric field, dielectric behaves like a conductor.