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Voltage Across A Capacitor

The voltage across a capacitor at any given time is determined by the history of the current that has flowed through it. Here's a plain English description of the formula:

Formula

The voltage across a capacitor at any given time is determined by the history of the current that has flowed through it. We can express this as the sum of the initial voltage plus a integral of the current over time.

\[ v(t) = v(t_0) + \frac{1}{C} \int_{t_0}^{t} i(t)dt \]

where:

  • \(v(t)\): Voltage across the capacitor at time \(t\)
  • \(v(t_0)\): Initial voltage across the capacitor at time \(t_0​\)
  • \(C\): Capacitance (in farads)
  • \(i(t)\): Current flowing through the capacitor at time \(t\)
  • \(t_0​\): Initial time (often taken as t=0)

Interactive Simulation

Below is a p5.js sketch that demonstrates the voltage across a capacitor over time in an RC (resistor-capacitor) circuit. The simulation allows you to adjust the resistance (R) and capacitance (C) values using sliders and observe how they affect the charging and discharging curves of the capacitor.

Run the Voltage Across a Capacitor Simulation

Description:

  • Sliders: Adjust the values of Resistance (R) and Capacitance (C).
  • Buttons: Choose between charging and discharging the capacitor.
  • Graph: Displays the voltage across the capacitor over time.
  • Real-time Voltage: Shows the instantaneous voltage value.

Understanding Voltage Across a Capacitor

  1. Basic Concept: A capacitor stores electrical energy in the form of an electric field created by a separation of charges. The voltage across the capacitor reflects the amount of charge stored relative to its capacity to store charge (capacitance).

  2. Relationship with Current:

    • Current Flow: When current flows into the capacitor, it accumulates charge on the plates, increasing the voltage across it.
    • Accumulated Charge: The total charge accumulated over time affects the voltage. This accumulation is essentially the sum of all the small amounts of charge added over time.

    • Dependence on Capacitance:

    • Capacitance (CCC): This is a measure of the capacitor's ability to store charge per unit voltage. A larger capacitance means the capacitor can store more charge for the same voltage.

    • Effect on Voltage: For a given amount of accumulated charge, a capacitor with a larger capacitance will have a lower voltage across it compared to one with a smaller capacitance.

    • Initial Conditions:

    • Starting Voltage: If the capacitor already has an initial voltage at the starting time (often denoted as t0t_0t0​), this initial voltage must be considered in determining the voltage at a later time.

    • Continuity: The voltage across a capacitor cannot change instantaneously; it changes gradually as the charge accumulates or dissipates over time.

    • Time Dependency:

    • Integration Over Time: To find the voltage at a specific time, you consider all the current that has flowed into the capacitor from the starting time to that specific time.

    • Sum of Contributions: Each small interval of current flow contributes to the total charge, and thus the voltage, at the time you're interested in.

    • Practical Interpretation:

    • Charging: When you apply a current to a capacitor, the voltage rises as the capacitor charges up.

    • Discharging: If the current is flowing out of the capacitor (negative current), the voltage decreases as the capacitor discharges.
    • Circuit Behavior: In a circuit, this relationship explains how capacitors smooth out voltage changes and can filter signals.

Summary

  • Voltage Across Capacitor: Determined by the accumulated charge divided by the capacitance, plus any initial voltage present.
  • Accumulated Charge: Found by summing (integrating) the current over time from the starting point to the time of interest.
  • Capacitance Effect: Higher capacitance means more charge is needed to achieve the same voltage.
  • Initial Voltage: Any voltage present at the starting time must be included in the total voltage calculation.
  • Time Consideration: The voltage change is continuous and depends on the entire history of current flow up to that moment.

Real-World Example

Imagine filling a bucket (capacitor) with water (charge) using a hose (current):

  • Flow Rate: The rate at which water flows from the hose is like the current.
  • Bucket Size: The size of the bucket represents the capacitance.
  • Water Level: The level of water in the bucket corresponds to the voltage across the capacitor.
  • Filling Over Time: The total amount of water in the bucket at any time depends on how long and how fast water has been flowing into it.
  • Starting Level: If the bucket already had some water at the beginning, this initial amount affects the current water level.

By understanding these concepts, you can grasp how the voltage across a capacitor evolves over time based on the current flowing into or out of it and the capacitor's properties, all without needing to refer to the mathematical equation itself.