Capacitor Lab: Electric Fields & Energy Storage

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Introduction

A 100 μF capacitor rated at 25V shouldn't hurt you. The datasheet says it stores about 31 millijoules. But probe one that's been sitting on a bench after a power supply test, and you'll discover that "discharged" is a relative term. The leakage current that slowly drained it? That same high impedance path makes it drain slowly when you think it's safe. Experienced engineers keep a bleeder resistor handy. Everyone else learns the hard way.

In an ideal world, capacitors would be simple: two plates, a dielectric, predictable behavior. Real circuits tell a different story. That ceramic bypass cap on your microcontroller? Its capacitance drops 50% at DC bias voltage [1]. The electrolytic in your audio amp? It's also a resistor, an inductor, and occasionally an explosive device if you reverse the polarity. When you probe a node with your oscilloscope, you're adding 10-15 pF of parasitic capacitance that can completely change high-frequency behavior.

Our simulation lets you explore the ideal physics (where C = κε₀A/d actually holds) while building intuition for where real circuits diverge. You'll manipulate geometry, swap dielectrics, watch fields form, and run RC transients. Most simulators stop there. Ours includes a field detector you can drag around the plates, measuring local E-field strength at any point. That feature addresses a gap we noticed in other tools: they show the field exists but don't let you quantify it spatially.

How to Use This Simulation

In an ideal world, you'd trust C = kappa times epsilon-zero times A over d, plug in values, and be done. But real circuits have parasitic capacitances that add up, dielectric absorption that "remembers" previous voltages, and breakdown voltages that end experiments abruptly. This simulator helps you build intuition for ideal behavior before real-world parasitics complicate your design.

Visualization Controls

Input Parameters

Material Presets

Output Display

The results panel shows calculated values in real-time:

• Capacitance: C in farads (with auto-scaling to pF, nF, uF)
• Charge: Q = CV on the plates
• E-Field: V/d between plates (uniform in ideal case)
• Energy: U = 0.5 times C times V-squared
• Density: Energy per unit volume of dielectric
• Status: Safe, Warning (near breakdown), or Danger (exceeds air breakdown)

The energy bar shows stored energy relative to the maximum possible with current parameters.

Field Detector

Click anywhere on the capacitor canvas to measure local field values. The signal sees:

• E at point: Local electric field strength (V/m)
• V at point: Potential at that location (interpolated between plates)

Between the plates, E is uniform and V drops linearly. Outside the plates, both values approach the plate values. This helps visualize fringe effects at the edges.

RC Circuit Panel

The right panel simulates charging and discharging behavior:

• Charge button: Starts exponential rise to supply voltage
• Discharge button: Starts exponential decay to zero
• Reset button: Clears the RC history and restores initial state
• tau display: Shows RC time constant in milliseconds
• V(t) display: Real-time capacitor voltage during transients

Tips for Exploration

When you probe this node on real capacitors, your scope adds 10-15 pF of probe capacitance. The signal sees this as extra C in parallel. Use these exercises to understand the relationships:

1. Double the plate area and observe capacitance double. Then halve the separation and watch it double again. The formula C = kappa times epsilon-zero times A over d becomes intuitive.

2. Switch from Air to Ceramic preset and observe the 6.5x increase in capacitance. This is why ceramic caps dominate modern electronics despite their voltage coefficient issues.

3. Increase voltage while watching the Status indicator. At high V/d ratios, you approach air breakdown (3 MV/m). Real capacitors fail dramatically at these field strengths.

4. Run the RC Charge cycle and watch the exponential curve. After 1 tau, voltage reaches 63.2% (not 50%). After 5 tau, it is 99.3% charged. This is the universal RC behavior.

5. Click inside and outside the plates with the field detector. Inside shows uniform E = V/d. Outside shows fringe effects and reduced field strength.

6. Compare energy at 10V vs 20V. Energy quadruples because U = 0.5 times C times V-squared. This is why high-voltage capacitor banks demand respect.

What Is a Capacitor?

A capacitor stores energy in an electric field between two conductors separated by an insulator [1]. The signal sees this as a frequency-dependent impedance: open circuit at DC, increasingly conductive at higher frequencies. That dual nature makes capacitors indispensable for filtering, coupling, and energy storage.

The governing equation is:

Q = CV

Charge (Q) equals capacitance (C) times voltage (V). Simple enough in textbooks.

Capacitance depends on geometry [2]:

C = κε₀A/d

Where κ is the dielectric constant, ε₀ = 8.854 × 10⁻¹² F/m (permittivity of free space), A is plate area, and d is separation. Double the area, double the capacitance. Halve the gap, double it again. In practice, you'll find that separation has manufacturing limits (dielectric breakdown), and area means size and cost.

How the Simulator Works

The field detector (click anywhere on the canvas) reports local E-field magnitude and potential. This is the feature that most other capacitor simulations lack, and practitioners tell us it's what they actually wanted for teaching fringe effects and field uniformity.

Types of Capacitors

Parallel Plate Capacitors

The textbook case. Two flat conductors, uniform field between them (mostly). Variable tuning caps in old radios used this geometry. You'll rarely see discrete parallel plate caps in modern designs, but the physics underlies everything else.

Ceramic Capacitors (κ = 6 to 10,000+)

The workhorse. X7R, X5R, C0G/NP0 designations specify temperature coefficients. The gotcha: high-K ceramics (X7R, X5R) lose capacitance under DC bias. A "10 μF" cap might give you 5 μF at its rated voltage [3]. C0G maintains capacitance but maxes out around 10 nF in practical sizes.

Electrolytic Capacitors

Enormous capacitance per volume. The dielectric is an oxide layer only nanometers thick (hence large C/d ratio). Polarity matters: reverse it and the oxide reforms on the wrong electrode, generating gas until the can ruptures. Experienced engineers have seen this happen. You don't want to.

Film Capacitors

Polyester, polypropylene, polystyrene films. Stable, low losses, no polarity concerns. Audio crossover networks, motor run applications, precision timing. More expensive per microfarad than electrolytics, worth it when stability matters.

Supercapacitors (Electric Double-Layer)

Capacitance measured in Farads, not microfarads. Store energy like a battery but charge/discharge like a capacitor. Lower energy density than batteries but handle many more cycles and deliver power in bursts. You'll find them in regenerative braking systems, backup power, and anywhere you need to absorb or deliver current spikes.

Key Parameters

Dielectric Material Properties

Essential Formulas

Capacitance

C = κε₀A/d

Larger plates, smaller gap, higher dielectric constant: more capacitance.

Stored Charge

Q = CV

Charge accumulates proportionally to applied voltage.

Electric Field (Uniform Region)

E = V/d = σ/(ε₀κ)

Between ideal parallel plates, the field is uniform. Our field detector shows you where this approximation holds and where fringe effects appear.

Stored Energy

U = ½CV² = ½QV = Q²/(2C)

The V² term matters. Double the voltage, quadruple the energy. This is why 400V capacitor banks in power supplies demand respect.

Energy Density

u = ½ε₀κE²

Energy per unit volume scales with field strength squared.

RC Time Constant

τ = RC

After one time constant, voltage reaches 63.2% of final value (not 50%, not 67%). After five time constants, you're at 99.3%, close enough to "done" for most purposes [4].

Charging Transient

V(t) = V₀(1 - e^(-t/τ))

Discharging Transient

V(t) = V₀e^(-t/τ)

Learning Objectives

After working through this simulation, you will be able to:

1. Calculate capacitance from geometry using C = κε₀A/d and predict how changes affect the result
2. Measure electric fields spatially using the field detector to understand uniformity and fringe effects
3. Compare dielectrics and explain why material choice determines both capacitance and voltage limits
4. Analyze RC transients by computing time constants and predicting voltage curves
5. Evaluate energy storage and explain why the V² relationship has safety implications
6. Identify breakdown risk by comparing calculated field strength to material limits

Exploration Activities

Activity 1: Geometry Dependence

Objective: Verify C = κε₀A/d holds in simulation

Steps:

1. Set air dielectric (κ = 1), V = 12V, A = 100 cm², d = 1.0 mm
2. Record the capacitance value
3. Double the plate area to 200 cm², observe capacitance doubles
4. Return to 100 cm², halve separation to 0.5 mm, observe capacitance doubles again
5. Confirm both changes are predicted by the formula

What to observe: Capacitance responds linearly to area and inversely to separation. No surprises here, which is exactly the point.

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Activity 2: Dielectric Comparison

Objective: Understand why material choice matters

Steps:

1. Start with air (κ = 1), note capacitance
2. Switch to mica (κ = 5.5), observe the increase
3. Switch to ceramic (κ = 6.5), observe further increase
4. Note that higher κ also means lower breakdown strength

What to observe: The formula works, but real designs trade κ against breakdown margin. A ceramic cap with 6.5× the capacitance might handle only 1/3 the voltage before dielectric failure.

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Activity 3: Energy and Voltage

Objective: Experience why voltage ratings matter

Steps:

1. Set V = 10V, record stored energy
2. Set V = 20V, record stored energy
3. Calculate the ratio (should be 4:1, not 2:1)
4. Consider what happens at V = 100V (100× the energy of V = 10V)

What to observe: Doubling voltage quadruples stored energy. This is why capacitor failures can be violent at high voltages.

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Activity 4: RC Charging Dynamics

Objective: Understand transient behavior

Steps:

1. Set capacitance around 100 pF, R = 1 kΩ
2. Calculate τ = RC (should be around 0.1 μs)
3. Click "Charge" and watch the exponential rise
4. Note voltage at t = τ (63.2% of supply)
5. Note that 5τ reaches 99.3%

What to observe: The exponential shape is universal for RC circuits. Changing R or C just rescales time.

Real-World Applications

1. Power Supply Decoupling: Every IC needs bypass caps close to power pins. When the chip switches, it draws current spikes. The capacitor supplies that current locally while the power supply catches up through its inductance. Too small a cap means voltage droop. Too far away means the inductance of the trace defeats the purpose.

2. Sample-and-Hold Circuits: ADCs sample analog signals onto capacitors, then convert the held voltage. Leakage current and dielectric absorption (the cap "remembering" previous voltages) limit accuracy. Polypropylene film caps minimize these effects.

3. Snubber Networks: When you open a switch carrying inductive load current, voltage spikes can arc across contacts or destroy semiconductors. An RC snubber absorbs the energy. The capacitor takes the initial surge; the resistor damps the subsequent ring.

4. Touch Sensing: Your finger is a conductor with some capacitance to ground. Bring it near a sense electrode and the total capacitance changes. Controllers detect these changes at the picofarad level, enabling the touch screens on every smartphone.

5. Defibrillators: Charge a large capacitor to hundreds of volts over several seconds, discharge through paddles in milliseconds. The capacitor handles the power density that batteries cannot provide directly.

Common Misconceptions

1. Capacitance Changes with Voltage

Capacitance is a geometric property (for ideal capacitors). Charge changes with voltage (Q = CV), but C stays constant. Except: high-K ceramics exhibit voltage coefficient, meaning C actually does decrease at higher DC bias [3]. The datasheet curve you need is often on page 3 in tiny print.

2. 50% Charge at One Time Constant

It's 63.2%, not 50%. The factor (1 - 1/e) appears because charging follows an exponential, not linear, curve. After 5τ, you reach 99.3%. Calling it "fully charged" at that point is an engineering judgment, not a mathematical truth.

3. More Capacitance Is Always Better

A larger cap means slower transient response in RC circuits, lower self-resonant frequency, and higher cost. Bypass caps actually stop working above their resonant frequency because internal inductance takes over. Sometimes a 100 nF and 100 pF in parallel outperform a single 100 nF.

4. Electrolytic Polarity Doesn't Matter at Low Voltage

It always matters. The oxide dielectric forms on one plate. Reverse voltage, even briefly, and you start forming oxide on the other plate while stripping the original. Gas generation follows. Case rupture follows that. Don't test this.

Frequently Asked Questions

Why does capacitance increase when I insert a dielectric?

The dielectric polarizes in the field, creating internal dipoles that partially cancel the external field from the plates [1]. To maintain the same voltage (same field times distance), the capacitor must store more charge. Same voltage, more charge: higher capacitance.

What happens to stored energy when I insert a dielectric into a charged, isolated capacitor?

Energy decreases. Charge Q is constant (isolated), but C increases, so U = Q²/(2C) drops [2]. The "missing" energy went into mechanical work pulling the dielectric into the field (polarized materials are attracted toward stronger fields).

Why are practical capacitors measured in μF or pF instead of Farads?

One Farad is enormous. A parallel plate capacitor with 1 mm separation would need plates covering about 100 km² to reach 1 F [5]. Supercapacitors achieve Farads through electrode surface areas measured in thousands of square meters per gram (nanoporous carbon).

How do I select a capacitor for a specific application?

Consider: (1) Required capacitance, (2) Voltage rating with margin (typically 50% above expected max), (3) ESR for power handling, (4) Temperature coefficient for precision apps, (5) Size and mounting, (6) Polarity requirements, (7) Frequency range versus self-resonant frequency [4].

What limits how fast a capacitor can charge?

Every circuit has resistance: wire, source impedance, intentional series resistance. That creates an RC time constant. Even with zero resistance, inductance creates LC resonance that limits rise time. Nothing charges instantly in real circuits.

References

1. MIT OpenCourseWare 8.02: Electricity and Magnetism lecture notes on capacitance and dielectrics. Available at: https://ocw.mit.edu/courses/8-02-physics-ii-electricity-and-magnetism-spring-2007/ (CC BY-NC-SA)

2. HyperPhysics - Capacitance: Georgia State University resource on capacitor physics. Available at: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capac.html (Free educational)

3. Murata - MLCC Characteristics: Technical note on ceramic capacitor DC bias effects. Available at: https://www.murata.com/en-us/products/capacitor/mlcc/basic/characteristics (Free technical resource)

4. All About Circuits - Capacitors: Engineering resource on selection and design. Available at: https://www.allaboutcircuits.com/textbook/direct-current/chpt-13/capacitors-and-capacitance/ (Free educational)

5. Electronics Tutorials - Capacitors: Practical capacitor types and applications. Available at: https://www.electronics-tutorials.ws/capacitor/cap_1.html (Free educational)

6. Engineering Toolbox - Capacitor Energy: Practical calculations. Available at: https://www.engineeringtoolbox.com/capacitor-energy-d_1233.html (Free educational)

7. LibreTexts Physics - Capacitance: Open-access textbook. Available at: https://phys.libretexts.org/Bookshelves/University_Physics (CC BY)

8. NIST CODATA: Physical constants reference. Available at: https://physics.nist.gov/cuu/Constants/ (Public domain)

About the Data

Physical constants (ε₀ = 8.854 × 10⁻¹² F/m) are from NIST CODATA values. Dielectric constants and breakdown strengths are representative engineering handbook values; actual values vary by material grade, purity, temperature, and frequency. The simulation implements an ideal parallel plate model assuming uniform fields and neglecting fringe effects at edges.

How to Cite

Simulations4All Team. "Capacitor Lab: Electric Fields & Energy Storage." Simulations4All, 2025. Available at: https://simulations4all.com/simulations/capacitor-lab

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