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Premium experiment • exponential transients

PHYSICS • ELECTRICITY

RC Circuit

Watch an ideal capacitor charge or discharge exponentially and connect the curve to the time constant τ = RC.

● INTERACTIVERC Circuit

Time constant

Capacitor voltage

Current

Stored energy

Reasoning inspector

live model state

Time constant τ—

Physical time—

Capacitor voltage—

Instantaneous current—

Initial current—

Capacitor charge—

Stored energy—

Transient progress—

Mode—

Explore first-order circuit transients

Change resistance and capacitance to see how the time constant controls the pace of charging and discharging.

LEARN WITH THE SIMULATION

Understand RC Circuit by experimenting.

START HERE

An RC circuit has one natural time scale: the time constant τ = RC. Instead of treating charging and discharging as two unrelated formulas, this laboratory makes their shared exponential structure visible. Resistance and capacitance set how many physical seconds one τ lasts, while the elapsed-time control is expressed in multiples of τ so the same 63.2%, 86.5% and 95.0% progress landmarks can be tested across very different component values.

BUILD THE IDEA

What you should understand first

Compute τ = RC and distinguish physical elapsed time t from the dimensionless ratio t/τ.
Explain why a charging capacitor reaches about 63.2% of its final voltage after one time constant while a discharging capacitor has about 36.8% remaining.
Track voltage, current, charge and stored electric energy during the same transient.
Predict which changes alter the speed in seconds and which leave the normalized exponential progress unchanged.

TRY IT YOURSELF

Why can two RC circuits with very different resistor and capacitor values be at the same percentage of completion?

Set charging mode and elapsed time to 1τ; record voltage progress and physical time.
Change resistance while keeping the elapsed-time slider at 1τ and compare the percentage progress with the new physical time.
Change capacitance and repeat.
Switch to discharge at 1τ and connect the remaining voltage fraction to e⁻¹.

Read the visual:

Normalized progress depends on t/τ, not on R or C separately. Changing R or C changes τ and therefore the number of seconds represented by one time constant, but the exponential fraction at a fixed multiple of τ stays the same.

WHY IT WORKS 01

The time constant separates circuit scale from curve shape

Multiplying resistance in ohms by capacitance in farads produces seconds, giving τ = RC. If time is rewritten as the dimensionless variable q = t/τ, the charging law becomes Vc/V = 1 − e⁻q and the discharging law becomes Vc/V0 = e⁻q. The curve shape therefore depends only on q, while R and C decide how quickly that dimensionless progress unfolds in real time.

WHY IT WORKS 02

Current is largest when the voltage difference is largest

At the start of charging, the capacitor voltage is zero in this model, so the resistor initially sees the full source voltage and the current magnitude is V/R. As charge accumulates, Vc approaches the source voltage, the resistor voltage shrinks and current decays exponentially. During discharge the current reverses sign because stored capacitor energy drives charge in the opposite direction through the resistor.

WHY IT WORKS 03

Energy follows voltage squared, not voltage linearly

The electric energy stored in an ideal capacitor is E = ½CVc². That squared dependence means a capacitor at half its final voltage stores only one quarter of its final energy. The inspector exposes voltage, charge Q = CV and energy simultaneously, making it possible to distinguish three quantities that are often treated as if they grew in the same proportion.

DO NOT MIX THESE UP

Common misconception

A capacitor does not become mathematically 100% charged after one time constant; at 1τ it is only about 63.2% of the way to its final voltage. Five time constants is a practical engineering approximation to completion, not a discontinuous cutoff. The ideal model also assumes a constant source and ideal resistor and capacitor, with no leakage, ESR or source resistance.

PRACTICE CHALLENGE

Can you hit a target transient progress and then predict the physical time for a different R or C?

Choose a target progress in the premium controls and adjust elapsed time in units of τ until it is reached.
Record the corresponding t/τ and physical time.
Change resistance or capacitance without changing the normalized elapsed-time setting.
Predict the new physical time before reading the inspector, and verify that the normalized progress is unchanged.

Reveal the reasoning

For an ideal first-order RC transient, the fractional progress is controlled by q = t/τ. Once a target fraction fixes q, the required physical time is simply t = qRC. Changing R or C scales the required number of seconds in direct proportion while leaving the normalized exponential fraction unchanged.

MODEL BOUNDARIES

What this simulation assumes

Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.

Model assumptions

The resistor and capacitor are ideal lumped linear components with constant values.
The voltage source and switching connection are ideal with zero internal or wire resistance.
The circuit follows the standard first-order exponential response of a single series RC network.

Model limitations

Capacitor leakage, equivalent series resistance, parasitic inductance and source impedance are omitted.
The model is not intended for nonlinear capacitors, dielectric breakdown or high-frequency distributed circuit behavior.

CONNECT IT

Where this matters

Timing and delay networks
Sensor smoothing and low-pass circuits
Capacitor charging and discharge design
Understanding first-order electrical transients

WORDS THAT MATTER

Key vocabulary

Time constant: Characteristic time τ = RC that sets the physical time scale of an ideal first-order RC transient.
Charging: Transient in which capacitor voltage approaches the source voltage as V(1−e⁻ᵗ/τ).
Discharging: Transient in which capacitor voltage decays from its initial value as V0e⁻ᵗ/τ.
Capacitor charge: Stored electric charge Q = CV across an ideal capacitor.
Stored energy: Electric potential energy E = ½CV² held by an ideal capacitor.

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