Many electronics devices can be dangerous even after they...

Many electronics devices can be dangerous even after they are shut off. Consider an $\mathrm{RC}$ circuit with a $150 .-\mu \mathrm{F}$ capacitor and a $1.00-\mathrm{M} \Omega$ resistor connected to a 200.-V power source for a long time and then disconnected and shorted, as shown in the figure. How long will it be until the potential difference across the capacitor drops to below $50.0 \mathrm{~V} ?$

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Key Concepts

Exponential Decay

When a capacitor discharges through a resistor in an RC circuit, the voltage across the capacitor decreases exponentially with time. This decay is described by the equation V(t) = V0 exp(?t/?), where V0 is the initial voltage and ? is the time constant. Understanding this exponential behavior is essential for calculating the time it takes for the voltage to drop to a certain level, which is crucial in both design and safety assessments.

RC Circuit

An RC circuit consists of a resistor and a capacitor connected together. It is fundamental in understanding how circuits respond over time, especially in processes related to charging and discharging capacitors. The interplay between resistance and capacitance in such circuits dictates the behavior of transient responses, which are critical in timing and filtering applications.

Time Constant

The time constant, usually denoted by ? (tau), is the product of the resistance (R) and the capacitance (C) in an RC circuit. It provides a measure of the time required for the capacitor to charge up to about 63% of its maximum voltage or to discharge to about 37% of its initial voltage, serving as a key parameter in predicting how quickly the circuit responds.

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