Discharging A Capacitor (Copy)

Key Concepts of Capacitor Discharge

• A charged capacitor discharges when connected across a resistor.
• During discharge:
  • The charge (Q) on the capacitor decreases
  • The current (I) in the circuit decreases
  • The potential difference (V) across the capacitor decreases
• The variation of these quantities with time follows an exponential decay.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Time Constant (τ = RC)

• The time constant (τ) is a measure of how quickly the capacitor discharges.
• τ = RC
  • R = resistance in ohms (Ω)
  • C = capacitance in farads (F)
• After a time equal to τ:
  • Q, V, and I all decrease to approximately 37% (i.e. 1/e) of their initial values.
• The larger the time constant, the slower the discharge.

Exponential Decay Equations

• The following equations describe the decay of charge, current, and potential difference during discharge:
  • Q = Q₀ e^(–t/RC)
  • V = V₀ e^(–t/RC)
  • I = I₀ e^(–t/RC)

  Where:

  • Q₀, V₀, I₀ are the initial values
  • t is the time elapsed since discharge began
  • e is the base of natural logarithms ≈ 2.718

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Behaviour of Graphs

• Charge-Time (Q–t) Graph:
  • Exponential decay
  • Starts from Q₀ and decreases asymptotically toward 0
• Voltage-Time (V–t) Graph:
  • Same shape as Q–t graph
  • V ∝ Q, so identical behaviour
• Current-Time (I–t) Graph:
  • Also exponential decay
  • Starts from I₀ and decreases to zero
• After 5τ, the values of Q, V, and I are effectively zero (less than 1% of initial value)

Calculating Discharge Values

• At time t = 0:
  • Q = Q₀
  • V = V₀
  • I = I₀
• At time t = τ (RC):
  • Q = 0.37 Q₀
  • V = 0.37 V₀
  • I = 0.37 I₀
• At t = 2τ:
  • Q ≈ 0.14 Q₀
  • V ≈ 0.14 V₀
  • I ≈ 0.14 I₀

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Deriving the Equations (Overview)

• Applying Kirchhoff’s loop rule for a discharging capacitor:

  V = IR

  Q = CV → V = Q/C

  So:

  Q/C = IR = (dQ/dt) × R

  Rearranging gives a first-order differential equation:

  dQ/dt = –Q/RC

• Solving this gives the exponential decay law:

  Q = Q₀ e^(–t/RC)

Understanding the Exponential Behaviour

• The decay rate depends entirely on RC.
• This is not a linear decay — instead, the quantity decays faster at the beginning and slows down over time.
• This is typical for all first-order exponential decay systems.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Energy Dissipation During Discharge

• The energy stored in the capacitor is converted to thermal energy in the resistor.
• No energy is lost in an ideal capacitor, but in real circuits:
  • All stored energy is dissipated as heat in the resistor.
• Energy stored at the beginning:

  W = ½CV₀²

• As the capacitor discharges, this energy decreases over time.

Summary Table

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course