What is the time constant for the discharging of the...

What is the time constant for the discharging of the capacitors in the circuit shown in the figure? If the $2.00-\mu \mathrm{F}$ capacitor initially has a potential difference of $10.0 \mathrm{~V}$ across its plates, how much charge is left on it after the switch has been closed for a time equal to half of the time constant?

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Key Concepts

Time Constant

The time constant, usually denoted by ?, is a measure in RC circuits that quantifies the rate at which capacitors charge or discharge. It is given by the product of the resistance (R) and the capacitance (C) in the circuit, ? = RC. This value indicates the time required for the system to reach approximately 63.2% of its final value during a charging process or to drop to about 36.8% of its initial value during discharge.

Exponential Decay in RC Circuits

During the discharging process in an RC circuit, the charge on the capacitor decreases exponentially with time according to the formula Q = Q?e^(?t/?), where Q? is the initial charge. This exponential decay describes how the capacitor releases its stored energy, with the time constant ? determining the rate at which the decay occurs.

RC Circuit Analysis

RC circuits combine resistors and capacitors to form systems with transient behaviors that are analyzed using differential equations. Understanding the role of the time constant and the exponential nature of the charging and discharging processes is crucial for predicting how voltage and charge evolve over time in these circuits.

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