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The time constant, represented as \( \tau \), is a key concept when discussing the discharge of a capacitor through a resistor. It is defined as the time it takes for the charge remaining on a capacitor to decrease to approximately 37% of its initial value. Mathematically, it is given by the product of the resistance \( R \) and the capacitance \( C \), or \( \tau = RC \).

This value is critical because it determines the rate at which the capacitor discharges. A larger time constant means a slower discharge, while a smaller time constant results in a faster discharge. Understanding \( \tau \) helps predict how long it will take for the capacitor to lose a specific portion of its charge. It is important to note that the time constant is unique to each RC circuit combination, meaning different circuits will have different discharge rates based on their resistance and capacitance values.

In our exercise, knowing the time constant allows us to determine when the capacitor loses one-third or two-thirds of its charge. By linking \( \tau \) to these problems, we can form a clearer understanding of capacitor behavior during discharge.

Exponential decay is a crucial mathematical process that describes how a quantity decreases at a rate proportional to its current value. In the context of capacitor discharge, the quantity in question is the charge on the capacitor. The standard formula for exponential decay is given by \( q(t) = q_0 e^{-t/\tau} \), where \( q_0 \) is the initial charge, \( t \) is time, and \( \tau \) is the time constant.

One of the key features of exponential decay is that it is initially rapid and then gradually slows down. This means that in the early stages of discharge, the capacitor loses charge quickly. However, as time progresses, the rate of charge loss diminishes. This characteristic is important when calculating specific points like when one-third or two-thirds of the charge remains.

In the exercise, calculating the time to lose certain portions of charge involves solving the exponential decay equation for \( t \). This is achieved by considering natural logarithms, which help break down the equation into manageable steps, allowing us to find precise times for the charge levels mentioned.

Resistance plays a pivotal role in the context of circuits, affecting how quickly a capacitor discharges. In an RC (resistor-capacitor) circuit, the resistance \( R \) directly influences the time constant \( \tau \). This is because \( \tau \) is the product of resistance and capacitance, \( \tau = RC \).

The resistance opposes the flow of charge, thereby slowing down the discharge process as it increases. It works alongside the capacitance to determine how long it takes for the capacitor to return to equilibrium, or zero charge. Higher resistance means a larger time constant and thus a slower discharge. Conversely, lower resistance results in a smaller time constant, leading to a quicker discharge process.

In our exercise problem, understanding the interplay between resistance and capacitance allows us to accurately determine the time taken for specific charge loss events. The control that resistance provides over the discharge rate can be observed in how quickly a capacitor loses portions of its charge over time. This makes it an essential component in predicting the behavior of capacitors within various circuits.