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# The circuit elements of a simple defibrillator

## Problem

Defibrillators are medical devices used to treat life threatening cardiac arrhythmias. Defibrillators deliver electrical energy to the heart with the goal of depolarizing the heart muscle completely, and reverting the heart back to a normal sinus rhythm. A simple defibrillator consists of a power supply, a capacitor, an inductor, and a set of paddles. Figure 1 illustrates a circuit diagram of a simple defibrillator.
Figure 1: Schematic of a Simple Defibrillator
When the switch is connected to point A, the capacitor is connected to the power supply. The electrical current begins to flow through the circuit until the voltage at the terminals of the capacitor is equal to the voltage of the power supply. At this point, the capacitor has reached its total charging capacity and the current flowing through the circuit has stopped.
A graph depicting how the current changes with respect to time as the capacitor is being charged is illustrated below. Mathematically, the graph is represented by the equation I, equals, left parenthesis, V, start subscript, o, end subscript, slash, R, right parenthesis, e, start superscript, start text, left parenthesis, negative, t, slash, R, C, right parenthesis, end text, end superscript, where R, C is the product of the circuit resistance and the capacitor’s capacitance. R, C is also known as the time constant. The greater the time constant, the longer it takes for the current to approach zero.
Graph 1: Current vs Time graph for a Charging Capacitor
When the switch is connected to point B, the capacitor begins to discharge and electrical energy is delivered through the paddles to the heart. The current delivered to the heart must last for several milliseconds in order for the heart to completely depolarize. However, the current of a discharging capacitor decreases exponentially. An inductor is needed to prolong the duration of the current flow by inducing a voltage that opposes current flow. The tendency to resist current flow is called inductance. The amount of electrical energy delivered to the heart can be calculated by the equation: E, equals, left parenthesis, Q, times, V, start subscript, o, end subscript, right parenthesis, slash, 2 where Q is the charge on either plate of the capacitor and V, start subscript, o, end subscript is the voltage of the power supply.
Which of the following graphs correctly depicts how the charge (Q) on the capacitor varies with respect to time (t) when the capacitor is charging?