Capacitor Discharge

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Discharging a Capacitor

When a charged capacitor with capacitance C is connected to a resistor with resistance R, then the charge stored on the capacitor decreases exponentially.

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Discharge graph

  • Q=Q0etRCQ=Q_0e^{-\frac{t}{RC}}
  • Where Q0Q_0 is the initial charge on the capacitor.
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Time to halve

  • The time for the charge to reach half of the initial charge, T1/2T_{1/2}, can be found:
    • Q=Q02=Q0eT1/2RCQ=\frac{Q_0}{2}=Q_0e^{-\frac{T_{1/2}}{RC}}
    • 0.5=eT1/2RC0.5=e^{-\frac{T_{1/2}}{RC}}
    • ln(0.5)=ln(eT1/2RC)\ln (0.5)=\ln (e^{-\frac{T_{1/2}}{RC}})
    • ln(2)=T1/2RC-\ln(2)=-\frac{T_{1/2}}{RC}
    • T1/2=RCln2T_{1/2}=RC\ln 2
  • Because ln(2)=0.69, T1/2=0.69RC\ln(2) = 0.69,\space T_{1/2}=0.69RC.
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Time constant

  • The product RC is known as the time constant.
  • It is a property of exponential decay graphs that the curve will decrease by a constant fraction with each time constant.

Potential Difference and Current in a Discharging Capacitor

The potential difference and the current in a discharging capacitor have similar forms.

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Potential difference

  • The potential difference across a capacitor with time constant RC, initial potential difference V0, and having been discharged for a time t is:
    • V=V0etRCV=V_0e^{-\frac{t}{RC}}
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  • The current passing through a capacitor with time constant RC, initial current I0, and having been discharged for a time t is:
    • I=I0etRCI=I_0e^{-\frac{t}{RC}}
    • R is the resistance across the capacitor
  • I0=V0RI_0=\frac{V_0}{R}

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