You're designing an external defibrillator that discharges a...

You're designing an external defibrillator that discharges a capacitor through the patient's body, providing a pulse that stops ventricular fibrillation. Specifications call for a capacitor storing 250 J of energy; when discharged through a body with $40-\Omega$ transthoracic resistance, the capacitor voltage is to drop to half its initial value in $10 \mathrm{ms}$. Determine the capacitance (to the nearest $10 \mu \mathrm{F})$ and initial capacitor voltage (to the nearest $100 \mathrm{V}$ ) that meet these specs.

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Key Concepts

Capacitor Energy Storage

The energy stored in a capacitor is given by the equation E = ½ CV². This relationship links the energy available in the capacitor to its capacitance and the voltage across it. In applications like external defibrillators, this concept is fundamental, as it allows one to design circuits that deliver the required energy in a controlled manner to achieve therapeutic effects.

RC Circuit Dynamics

RC circuit dynamics refer to the behavior of circuits that contain both resistors and capacitors. The key parameter in such circuits is the time constant, ? = RC, which determines the rate at which the voltage across the capacitor decays. Understanding these dynamics is essential when designing systems that require precise timing of voltage changes, such as in defibrillator pulses.

Exponential Decay in Capacitor Discharge

When a capacitor discharges through a resistor, the voltage decreases exponentially according to V(t) = V? exp(-t/RC). This exponential decay model allows one to relate the time taken for the voltage to drop to a certain fraction of its initial value with the values of resistance and capacitance. This concept is critical when specifying the timing characteristics for applications that demand controlled energy release.

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