Capacitors & RC Circuits

Intermediate

Visualize how capacitors charge and discharge in RC circuits with exponential behavior.

RC Circuit Theory

An RC circuit consists of a resistor (R) and capacitor (C) connected in series. When connected to a voltage source, the capacitor charges exponentially, and when disconnected, it discharges exponentially through the resistor.

Charging Equations:

Q(t) = Q₀(1 - e^(-t/τ))
V(t) = V₀(1 - e^(-t/τ))
I(t) = (V₀/R)e^(-t/τ)

Discharging Equations:

Q(t) = Q₀e^(-t/τ)
V(t) = V₀e^(-t/τ)
I(t) = -(V₀/R)e^(-t/τ)

Time Constant (τ):

τ = R × C
  • • At t = τ: 63.2% charged/discharged
  • • At t = 2τ: 86.5% charged/discharged
  • • At t = 3τ: 95.0% charged/discharged
  • • At t = 5τ: 99.3% charged/discharged (effectively complete)

Key Properties:

  • Capacitance (C) measures charge storage ability (Farads)
  • Larger τ means slower charging/discharging
  • Current is maximum at t=0, decreases exponentially
  • Energy stored: E = ½CV²

Interactive Simulation

RC Circuit Diagram
Circuit Parameters
Real-Time Measurements
Time
0.000s
τ = 100.000s
Voltage
0.00V
0.0% of max
Current
0.00mA
Idle
Energy
0.00μJ
Stored

💡 How to Use:

  • Adjust voltage, resistance, and capacitance using sliders
  • Click "Start Charging" to charge the capacitor
  • Click "Start Discharging" to discharge through resistor
  • Watch the animated current flow (yellow dots = charging, blue = discharging)
  • Observe exponential curves in voltage and current graphs
  • Time constant τ = RC determines how fast the process occurs
  • After 5τ, the capacitor is ~99% charged/discharged

Key Concepts

Exponential Behavior:

Both charging and discharging follow exponential curves, not linear. The rate of change is fastest at the beginning and slows down as the capacitor approaches its final state.

Time Constant Significance:

The time constant τ = RC determines how fast the circuit responds. A larger resistance or capacitance increases τ, making the process slower. After 5τ, the process is essentially complete (99.3%).

Current Direction:

During charging, current flows into the capacitor (positive). During discharging, current flows out of the capacitor (negative). The magnitude decreases exponentially in both cases.

Energy Conservation:

Energy is stored in the electric field between capacitor plates. During discharge, this energy is dissipated as heat in the resistor. Half the energy from the battery is always lost to the resistor during charging.

Real-World Applications

Timing Circuits

RC circuits create precise time delays in electronics, from simple timers to computer clock circuits

Filters

Low-pass and high-pass filters use RC circuits to block or pass specific frequencies

Flash Photography

Camera flashes use capacitors to store energy and release it quickly for bright light

Power Supplies

Smoothing capacitors in power supplies filter out voltage ripples for stable DC output

Touchscreens

Capacitive touchscreens detect finger position by measuring capacitance changes

Defibrillators

Medical defibrillators charge capacitors to deliver controlled electric shocks to the heart

Discussion

Have questions about RC circuits? Join the conversation below.

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