Capacitor - Geometry and Charge Storage
Capacitance depends on geometry: parallel plate C = ε₀εrA/d, cylindrical and spherical have different formulas. Charge Q = CV, energy E = ½CV². Reactance Xc = 1/(2πfC) for AC.
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Parallel plate C proportional to area, inversely to separation Dielectric εr increases capacitance by factor of εr Cylindrical and spherical have different geometry factors Reactance Xc decreases with frequency
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Why: Capacitors store energy in electric fields. Geometry determines capacitance—parallel plates, cylinders, and spheres have different formulas. Essential for circuit design, filters, and energy storage.
How: Parallel plate: C = ε₀εrA/d. Cylindrical: C = 2πε₀εrL/ln(b/a). Spherical: C = 4πε₀εrab/(b-a). Charge Q = CV. Energy E = ½CV² = ½QV.
Run the calculator when you are ready.
⚙️ Capacitor Parameters
Area (A)
Plate area
📚 What is a Capacitor?
A capacitor is a passive electronic component that stores electrical energy in an electric field. It consists of two conductive plates separated by a dielectric (insulating) material.
Where C is capacitance (Farads), ε₀ is vacuum permittivity (8.854×10⁻¹² F/m), εᵣ is relative permittivity, A is plate area (m²), and d is plate separation (m).
Capacitors are fundamental components in electronics, used for energy storage, filtering, timing circuits, power factor correction, and signal coupling.
📐 Key Formulas
Capacitance Formulas
Parallel Plate: C = ε₀εᵣA/d
Cylindrical: C = 2πε₀εᵣL/ln(r₂/r₁)
Spherical: C = 4πε₀εᵣr₁r₂/(r₂-r₁)
Charge & Energy
Charge: Q = CV
Energy: E = ½CV² = Q²/(2C)
Energy Density: u = ½εE²
AC Behavior
Reactance: Xc = 1/(2πfC)
Impedance: Z = -jXc
Phase: Current leads voltage by 90°
RC Circuits
Time Constant: τ = RC
Charging: V(t) = V₀(1-e^(-t/τ))
Discharging: V(t) = V₀e^(-t/τ)
🔧 How Capacitors Work
Energy Storage: When voltage is applied, electrons accumulate on one plate and are removed from the other, creating an electric field. The energy stored is proportional to the square of the voltage.
DC Behavior: Capacitors block DC current once charged. They act as open circuits in steady-state DC circuits.
AC Behavior: Capacitors allow AC current to pass, with reactance inversely proportional to frequency. Higher frequencies see lower reactance.
Charging/Discharging: In RC circuits, capacitors charge and discharge exponentially with time constant τ = RC. After 5τ, the capacitor is essentially fully charged/discharged.
Dielectric Effect: Dielectric materials increase capacitance by reducing the electric field between plates. The relative permittivity (εᵣ) determines the increase factor.
⏰ When to Use Capacitors
Timing Circuits
RC circuits use capacitors for delays, oscillators, and pulse generation. Time constant τ = RC determines timing.
Filtering
Low-pass filters block high frequencies. High-pass filters block DC. Band-pass filters combine both.
Power Factor Correction
Capacitors compensate for inductive loads, improving power factor and reducing utility penalties.
Energy Storage
Supercapacitors store energy for backup power, regenerative braking, and peak power delivery.
Coupling/Decoupling
AC coupling capacitors pass signals while blocking DC. Decoupling capacitors filter power supply noise.
Tuning Circuits
LC resonant circuits use capacitors with inductors for frequency selection in radios and oscillators.
📊 Sample Examples
Example 1: Timing Circuit (555 Timer)
A 555 timer uses R = 10 kΩ and C = 10 µF. Calculate the time constant and pulse width.
Example 2: Low-Pass Filter Design
Design a low-pass filter with cutoff frequency 1 kHz using C = 0.1 µF. Find required resistance.
Example 3: Energy Storage
A 1000 µF capacitor charged to 400 V stores how much energy?
Example 4: Power Factor Correction
A motor needs 53 Ω capacitive reactance at 50 Hz for PF correction. What capacitor value?
Example 5: Audio Coupling
An audio amplifier needs AC coupling with Xc = 159 Ω at 20 Hz. Calculate capacitor value.
🌐 Capacitor Geometries
Parallel Plate
Most common geometry. C = ε₀εᵣA/d
Used in: Standard capacitors, variable capacitors, MEMS devices
Cylindrical
Coaxial design. C = 2πε₀εᵣL/ln(r₂/r₁)
Used in: Coaxial cables, high-voltage capacitors, RF applications
Spherical
Concentric spheres. C = 4πε₀εᵣr₁r₂/(r₂-r₁)
Used in: High-voltage applications, theoretical models, Van de Graaff generators
🔬 Dielectric Materials
| Material | εᵣ | Breakdown (V/µm) | Applications |
|---|---|---|---|
| Vacuum | 1.0 | - | Reference |
| Air | 1.0006 | 3 | Variable capacitors |
| Paper | 2-4 | 15-20 | Old capacitors |
| Mica | 5-7 | 100-200 | High-frequency |
| Ceramic | 10-10,000 | 10-100 | General purpose |
| Tantalum | ~27 | ~50 | Small size, stable |
| Aluminum Oxide | ~9 | ~600 | Electrolytic |
❓ Frequently Asked Questions
Why does energy scale with V²?
Energy is the integral of charge times voltage. Since Q = CV, and we integrate from 0 to V, we get E = ½CV². The ½ factor comes from the linear charging process.
Why do capacitors block DC?
Once charged, a capacitor's plates have equal and opposite charges. No net current flows because the electric field prevents further charge movement. DC sees infinite impedance.
What happens at high frequencies?
Reactance Xc = 1/(2πfC) decreases with frequency. At very high frequencies, capacitors act like short circuits, passing AC signals easily.
Can capacitors store energy forever?
No. Real capacitors have leakage resistance that slowly discharges them. Supercapacitors can hold charge for days, but eventually self-discharge.
Why are capacitors in parallel additive?
Parallel capacitors share the same voltage but store separate charges. Total charge Q_total = Q₁ + Q₂ = C₁V + C₂V = (C₁ + C₂)V, so C_total = C₁ + C₂.
Why are capacitors in series like resistors in parallel?
Series capacitors share the same charge but have different voltages. V_total = V₁ + V₂ = Q/C₁ + Q/C₂ = Q(1/C₁ + 1/C₂), so 1/C_total = 1/C₁ + 1/C₂.
📚 Official Data Sources
Physical constants and standard reference data
Capacitor physics and formulas
Capacitor theory and applications
Circuit analysis and capacitor behavior
Comprehensive electronics education
⚠️ Disclaimer
This calculator is for educational and design purposes. Real capacitors have parasitic effects (ESR, ESL, leakage) that may affect performance. For critical applications in power electronics, RF circuits, or safety systems, consult manufacturer datasheets and verify calculations with experimental testing. Always follow proper safety procedures when working with charged capacitors.
For educational and informational purposes only. Verify with a qualified professional.
🔬 Physics Facts
Parallel plate capacitance C = ε₀εrA/d; ε₀ = 8.854×10⁻¹² F/m
— HyperPhysics
Cylindrical capacitors used in coaxial cables; C = 2πεL/ln(b/a)
— NIST
Spherical capacitor formula C = 4πε₀ab/(b-a) for concentric spheres
— Khan Academy
Dielectric constant εr of water ~80; air ~1; ceramics 100-10000
— NIST
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