A heart defibrillator being used on a patient has an R C...

A heart defibrillator being used on a patient has an $R C$ time constant of $10.0 \mathrm{ms}$ due to the resistance of the patient and the capacitance of the defibrillator. (a) If the defibrillator has a capacitance of $8.00 \mu \mathrm{F},$ what is the resistance of the path through the patient? (You may neglect the capacitance of the patient and the resistance of the defibrillator.) (b) If the initial voltage is $12.0 \mathrm{kV}$, how long does it take to decline to $6.00 \times 10^{2} \mathrm{V}$ ?

Key Concepts

RC Circuit

An RC circuit consists of a resistor (R) and a capacitor (C) connected together, allowing the study of transient electrical behavior when the circuit is charged or discharged. These circuits are fundamental in understanding how voltage and current change over time in various electronic and electrical systems.

Time Constant

The time constant, denoted as ? (tau), is defined as the product of the resistance and the capacitance (? = RC) in an RC circuit. It represents the characteristic time over which the voltage or current in the circuit decays to about 36.8% of its initial value in an exponential decay process. Understanding the time constant is crucial for analyzing the speed of response of the circuit.

Exponential Decay in RC Circuits

In an RC circuit, the voltage across the capacitor decays exponentially over time following the equation V(t) = V?e^(?t/RC), where V? is the initial voltage. This exponential decay behavior is a key concept in analyzing transient responses, enabling the calculation of the time required for the voltage to drop to a given level.

Determining Circuit Parameters

By utilizing the relationships inherent in the RC circuit, such as ? = RC for the time constant and the exponential decay equation for voltage, one can calculate unknown parameters such as the resistance when the capacitance and time constant are known, or the time required for the voltage to change to a particular value. This method of analysis is widely applicable in both theoretical studies and practical applications in electronics.

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