Voltage, Power, and Energy Storage in a Capacitor

Main ideas and concepts

• The video continues a prior discussion on inductors and now focuses on capacitors, showing that many governing equations are structurally similar to the inductor case.
• Goal: derive key relationships for a capacitor between voltage, current, power, and energy.
• Central physical intuition:
  • Voltage across a capacitor arises from charge accumulation on its plates.
  • Since current is charge flow rate, integrating current over time corresponds to accumulating charge, which produces voltage.
• The derivations emphasize:
  • “If you know one time-behavior (current or voltage), you can compute the other (and then power and energy).”

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Capacitor voltage from current (derivation)

Starting relation (capacitor constitutive equation)

• Current through a capacitor:
  • [ I = C \frac{dV}{dt} ]

Solve for voltage in terms of current

• Rearranged:

  • [ dV = \frac{I}{C}\, dt ]

• Integrate both sides to get voltage as a function of time:

  • [ V(t) = \frac{1}{C}\int_{0}^{t} I(\tau)\, d\tau + V(0) ]

  • The constant of integration corresponds to the initial voltage (V(0)).

Practical meaning

• If voltage behavior is given, you can compute current using the derivative form:

  • [ I = C \frac{dV}{dt} ]

• If current behavior is given, you recover voltage via the integral form above.

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Physical/unit insight for why voltage involves an integral

• Uses the unit interpretation of current:
  • 1 ampere = 1 coulomb/second
• Therefore:
  • Integrating current over time effectively adds up charge that flows into (and accumulates on) the capacitor plates.
• More time at nonzero current ⇒ more accumulated charge ⇒ higher voltage.

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Capacitor power (two equivalent forms)

General power identity

• Always true:

  • [ P = IV ]

• Substitute capacitor current:

  • [ I = C \frac{dV}{dt} ]

• Power in terms of voltage:

  • [ P(t) = C\, V(t)\, \frac{dV}{dt} ]

Sign interpretation

• Positive power: capacitor is absorbing/drawing power from the circuit.
• Negative power: capacitor is returning/delivering power to the circuit.

Turn around: power in terms of current

• Use the earlier voltage-from-current result:

  • [ V(t) = \frac{1}{C}\int_{0}^{t} I(\tau)\, d\tau + V(0) ]

• Substitute into (P = IV):

  • [ P(t) = I(t)\left(\frac{1}{C}\int_{0}^{t} I(\tau)\, d\tau + V(0)\right) ]

Practical meaning

• Depending on what you’re given:
  • If you know (V(t)) → compute (P(t)) using (C V \, dV/dt)
  • If you know (I(t)) → compute (V(t)) by integration, then compute (P(t))

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Energy stored in a capacitor

Use power as energy rate

• Since power is energy flow:
  • [ P = \frac{dW}{dt} ]

Substitute capacitor power expression

• Using (P = C V \frac{dV}{dt}):

  • [ \frac{dW}{dt} = C V \frac{dV}{dt} ]

• Multiply both sides by (dt):

  • [ dW = C V\, dV ]

• Integrate:

  • [ W = \int C V\, dV = \frac{1}{2} C V^2 ]

Final energy formula

• [ W = \frac{1}{2} C V^2 ]

• Units:

  • energy in joules
• Comparison to inductors:
  • Inductor energy was (\tfrac{1}{2} L I^2)
  • Capacitor energy mirrors the same structure:
    • inductor: depends on current
    • capacitor: depends on voltage

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Methodology / instruction-style steps (how to use the formulas)

To find capacitor voltage from current

• Use:

  • [ V(t) = \frac{1}{C}\int_{0}^{t} I(\tau)\, d\tau + V(0) ]

• Steps:

  • Determine (C)
  • Compute the integral of (I(t)) from (0) to (t)
  • Add the initial voltage (V(0))

To find capacitor power

• If given voltage:

  • [ P(t) = C\, V(t)\, \frac{dV}{dt} ]

• If given current:

  • First compute voltage:

    • [ V(t) = \frac{1}{C}\int_{0}^{t} I(\tau)\, d\tau + V(0) ]

  • Then compute power:

    • [ P(t) = I(t)\,V(t) ]

To find energy stored in the capacitor

• Use:

  • [ W = \frac{1}{2} C V^2 ]

• Steps:

  • Determine (C)
  • Use the capacitor voltage (V) (at the time of interest)
  • Substitute into (\tfrac{1}{2} C V^2)

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Closing context / roadmap

• Capacitors are emphasized as:
  • more common than inductors in typical circuits (smaller/compact physically)
• The next topics mentioned:
  • solving capacitor circuit problems,
  • then moving toward mixed arrangements (capacitors and inductors),
  • and eventually AC analysis.

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Speakers / sources featured

• No named individual speaker is identified in the subtitles.
• Video/host mentions: “Learn anything at mathandscience.com” (as a source/website), but no person’s name is provided.