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Capacitance equals charge divided by voltage

Solution

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Capacitance Equation

The basic capacitance relationship ties capacitance to charge and voltage. A capacitor stores electrical energy by holding opposite charges on two conductive plates separated by an insulator.

C = Q / V

Stored Energy (C, V)

Stored energy depends on capacitance and voltage squared. Doubling the voltage quadruples the stored energy.

U = ½CV²

Parallel Plate Capacitor

Two flat conducting plates of area A separated by distance d with a dielectric of permittivity ε form the simplest capacitor. Larger plates and thinner gaps increase capacitance.

C = εA / d

Cylindrical Capacitor

A cylindrical capacitor consists of two coaxial conductors used for coaxial cables and cylindrical sensor geometries.

C = 2πεL / ln(b/a)

How It Works

A capacitor stores electrical energy by holding opposite charges on two conductive plates separated by an insulator. The basic relationship C = Q/V ties capacitance to charge and voltage. For specific geometries, dedicated formulas (parallel plate, cylindrical) account for physical dimensions and dielectric properties. Three stored-energy forms (U = ½CV², U = Q²/2C, U = QV/2) let you solve with whichever pair of variables you know.

Example Problem

A 470 µF capacitor is charged to 25 V. How much energy does it store?

  1. Convert: C = 470 × 10⁻⁶ = 0.00047 F
  2. U = ½ × 0.00047 × 25² = ½ × 0.00047 × 625
  3. U = 0.147 J (about 147 mJ)

If the same capacitor held 0.01 coulombs of charge, the voltage would be Q/C = 0.01/0.00047 ≈ 21.3 V.

When to Use Each Variable

  • Solve for Capacitance (C = Q/V) — when you know the charge stored and the voltage across the capacitor, e.g., measuring capacitance from a charge-discharge test.
  • Solve for Stored Energy (U = 1/2 CV^2) — when you know the capacitance and voltage and need the energy stored, e.g., sizing a capacitor bank for a camera flash.
  • Solve for Parallel Plate Capacitance — when you know the plate area, gap distance, and dielectric permittivity, e.g., designing a MEMS sensor.
  • Solve for Cylindrical Capacitance — when you know the coaxial conductor dimensions and dielectric, e.g., calculating capacitance per meter of coaxial cable.

Key Concepts

Capacitance is the ability to store electric charge per unit voltage. Three equivalent energy formulas (U = 1/2 CV^2, U = Q^2/2C, U = QV/2) let you work with whichever pair of variables you know. Physical capacitance depends on geometry: for parallel plates, it scales with plate area and inversely with gap distance; for cylindrical capacitors, it scales with length and inversely with the logarithm of the radius ratio. The dielectric material between conductors multiplies capacitance by its relative permittivity.

Applications

  • Power electronics: sizing filter capacitors for DC power supplies to smooth voltage ripple
  • Energy storage: calculating the energy capacity of supercapacitor banks for regenerative braking systems
  • Sensor design: using parallel plate capacitance changes to detect displacement in MEMS accelerometers and pressure sensors
  • RF engineering: determining the capacitance of coaxial cables and transmission lines for impedance matching
  • Timing circuits: selecting capacitor values for RC time constants in oscillators and delay circuits

Common Mistakes

  • Forgetting to convert microfarads to farads — 1 uF = 1e-6 F; entering 470 instead of 0.00047 produces energy values that are a million times too large
  • Confusing series and parallel capacitor combinations — parallel capacitances add directly, but series reciprocals add; the total series capacitance is always less than the smallest individual capacitor
  • Using relative permittivity where absolute permittivity is needed — the parallel plate formula uses absolute permittivity (epsilon = epsilon_r times epsilon_0); omitting epsilon_0 gives values that are 8.85e-12 times too large
  • Assuming voltage rating equals maximum charge — exceeding the rated voltage can cause dielectric breakdown and capacitor failure even if the calculated energy seems reasonable

Frequently Asked Questions

What is the difference between a farad and a microfarad?

One farad is an enormous amount of capacitance. Most practical capacitors are measured in microfarads (1 µF = 10⁻⁶ F), nanofarads (10⁻⁹ F), or picofarads (10⁻¹² F). Supercapacitors can reach several thousand farads.

How do capacitors combine in series vs parallel?

In parallel, capacitances add directly: Ctotal = C1 + C2. In series, reciprocals add: 1/Ctotal = 1/C1 + 1/C2, so the total is always less than the smallest individual capacitor.

Why does stored energy depend on voltage squared?

As voltage rises, each additional increment of charge requires more work against the growing electric field. The result is a quadratic relationship: doubling the voltage quadruples the stored energy.

What is a parallel plate capacitor?

A parallel plate capacitor is two flat conductive plates separated by an insulating material (dielectric). Its capacitance depends on the plate area, the gap between plates, and the dielectric’s permittivity: C = εA/d. It is the most common theoretical model for understanding capacitance.

What are common uses of capacitors?

Capacitors are used for power supply filtering, signal coupling, timing circuits (RC), motor starting, camera flashes, defibrillators, and power factor correction. Decoupling caps near ICs stabilize voltage and reduce electromagnetic interference.

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