What is Capacitance?
Capacitance is the ability of a conductor or system of conductors to store electric charge per unit potential difference. A capacitor stores energy in an electric field between two conducting plates. It is measured in Farads (F) and defined by the formula C = Q / V.
Core Capacitance Formulas
| Symbol | Quantity | SI Unit | Notes |
|---|
| C | Capacitance | Farad (F) | 1 F = 1 C/V; practical units: μF, nF, pF |
| Q | Charge stored | Coulomb (C) | Q = C × V |
| V | Potential difference | Volt (V) | Across capacitor plates |
| U | Energy stored | Joule (J) | Stored in the electric field |
| R | Resistance | Ohm (Ω) | In series with capacitor in RC circuit |
| τ | Time constant | Second (s) | τ = RC; time to charge to ~63.2% |
| ε0 | Permittivity of free space | F/m | 8.854 × 10−12 F/m |
RC Circuit Theory & Key Concepts
The RC Time Constant (τ)
The RC time constant (τ = RC) quantifies how fast a capacitor charges or discharges. After one time constant, a charging capacitor reaches approximately 63.2% of the supply voltage. After 5τ, the capacitor is considered fully charged (99.3%). This is critical in designing filter circuits, timing circuits, oscillators, and pulse shaping networks.
After 1τ
Capacitor charges to 63.2% of V0 (charging) or discharges to 36.8% of V0 (discharging)
After 2τ
Charges to 86.5% (charging) or falls to 13.5% (discharging)
After 3τ
Charges to 95.0% — used as practical full-charge in many circuits
After 5τ
Charges to 99.3% — universally accepted as fully charged
Series and Parallel Combination of Capacitors
In a series combination, the same charge appears on each capacitor, but the voltage divides. In a parallel combination, all capacitors share the same voltage but charges add up. The parallel combination rule mirrors resistors in series, and vice versa.
Effect of Dielectric
When a dielectric material (insulator) of dielectric constant K (relative permittivity εr) is inserted between the plates, the capacitance increases by a factor of K:
C = K × ε0 × A ÷ d [where K = εr ≥ 1]
Common dielectrics: Air (K ≈ 1), Paper (K ≈ 3.7), Mica (K ≈ 5–8), Water (K ≈ 80), Barium Titanate (K ≈ 1200). A higher K allows more charge to be stored at the same voltage, which is why modern ceramic capacitors use barium titanate.
Solved Examples (NCERT / JEE Level)
Example 01 — Charge and Energy Stored
A 50 μF capacitor is connected across a 200 V DC supply. Calculate (a) the charge stored and (b) the energy stored in the capacitor.
Given: C = 50 μF = 50 × 10−6 F, V = 200 V
(a) Charge: Q = C × V
Q = 50 × 10−6 × 200
Q = 0.01 C = 10 mC
(b) Energy: U = (1/2) × C × V²
U = (1/2) × 50 × 10−6 × (200)²
U = (1/2) × 50 × 10−6 × 40000
U = (1/2) × 2.0
U = 1.0 J
Q = 10 mC | U = 1.0 Joule
Example 02 — RC Time Constant and Charging Voltage
An RC circuit has R = 10 kΩ and C = 100 μF connected to a 9 V supply. Find (a) the time constant and (b) the voltage across the capacitor after 1 second.
Given: R = 10 × 103 Ω, C = 100 × 10−6 F, V0 = 9 V, t = 1 s
(a) Time Constant: τ = R × C
τ = 10000 × 100 × 10−6
τ = 1.0 s
(b) Charging voltage: V(t) = V0 × (1 − e−t/τ)
V(1) = 9 × (1 − e−1/1)
V(1) = 9 × (1 − e−1)
V(1) = 9 × (1 − 0.3679)
V(1) = 9 × 0.6321
V(1) = 5.689 V
τ = 1.0 s | V(1s) = 5.689 V (63.21% of 9 V)
Example 03 — Series Combination of Capacitors
Three capacitors of capacitance 4 μF, 6 μF, and 12 μF are connected in series across a 120 V supply. Find the equivalent capacitance, total charge, and voltage across each capacitor.
Given: C1 = 4 μF, C2 = 6 μF, C3 = 12 μF, V = 120 V
Step 1: 1/Ceq = 1/4 + 1/6 + 1/12
= 3/12 + 2/12 + 1/12 = 6/12 = 1/2
Ceq = 2 μF
Step 2: Total charge Q = Ceq × V = 2 × 10−6 × 120 = 240 μC
Step 3: Voltage across each (same Q for series):
V1 = Q/C1 = 240/4 = 60 V
V2 = Q/C2 = 240/6 = 40 V
V3 = Q/C3 = 240/12 = 20 V
Ceq = 2 μF | Q = 240 μC | V1 = 60V, V2 = 40V, V3 = 20V | Check: 60+40+20 = 120 V ✓
Frequently Asked Questions (FAQs)
What is the SI unit of capacitance and why is the Farad so large?
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The SI unit of capacitance is the Farad (F), named after Michael Faraday. One Farad is an enormous capacitance — it means storing 1 Coulomb of charge at 1 Volt. In practice, most electronic capacitors range from picofarads (pF, 10−12 F) to microfarads (μF, 10−6 F). Only supercapacitors or ultracapacitors used in energy storage applications reach the Farad range.
What is the significance of the RC time constant?
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The RC time constant (τ = RC) tells you how quickly a capacitor charges or discharges through a resistor. After one time constant, the capacitor reaches 63.2% of the final voltage while charging (or drops to 36.8% while discharging). Engineers use τ to set the timing of oscillators, delay circuits, low-pass and high-pass filters, and camera flash circuits. A larger R or C gives a longer, slower response; smaller values give faster response.
Does a capacitor allow AC or DC current to pass through it?
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A capacitor blocks DC (direct current) in steady state because once fully charged, no current flows through it. However, it passes AC (alternating current) because the continuously changing voltage keeps charging and discharging the capacitor, allowing current to flow. This property makes capacitors essential as coupling capacitors (passing AC signals while blocking DC bias) and as bypass capacitors (filtering out AC noise from DC power lines).
How does inserting a dielectric increase capacitance?
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When a dielectric is placed between capacitor plates, its molecules become polarized in the electric field, creating an internal field that opposes the external field. This reduces the effective electric field and therefore the voltage across the capacitor for the same stored charge (Q). Since C = Q/V, a lower V with the same Q means higher capacitance. The factor by which capacitance increases is called the dielectric constant (K) or relative permittivity (εr).
What is the difference between capacitors in series and parallel?
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In a series combination, capacitors share the same charge (Q) but the voltage divides. The equivalent capacitance is always less than the smallest individual capacitance. Use series when you need a lower total capacitance or to handle a higher combined voltage rating. In a parallel combination, all capacitors share the same voltage but charges add. The equivalent capacitance is the sum of all individual values. Use parallel when you need a higher total capacitance.