A circuit contains a D cell battery, a switch, a 20-Ωresistor, and four 20-mF capacitors connect

step 1 For this question we are given a circuit that actually contains four capacitors connected in ser

A circuit contains a D cell battery, a switch, a...

A circuit contains a D cell battery, a switch, a $20-\Omega$ resistor, and four $20-\mathrm{mF}$ capacitors connected in series. (a) What is the equivalent capacitance of the circuit? (b) What is the $R C$ time constant? (c) How long before the current decreases to $50 \%$ of the initial value once the switch is closed?

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Key Concepts

Equivalent Capacitance

This concept involves calculating the total capacitance when multiple capacitors are connected in series. In series, the overall capacitance is determined by summing the reciprocals of the individual capacitances and then taking the reciprocal of that sum. This results in an equivalent capacitance that is always less than the smallest individual capacitor in the series.

RC Time Constant

The RC time constant is the product of the resistance (R) and the equivalent capacitance (C) in a circuit. It represents the characteristic time over which the capacitor charges or discharges. A larger time constant means the circuit responds more slowly, while a smaller time constant indicates a faster response.

Exponential Decay in RC Circuits

In RC circuits, both the voltage across the capacitor and the current in the circuit change exponentially over time when charging or discharging. The exponential decay is described by an equation of the form I(t) = I0 exp(-t/RC), where I0 is the initial current. This relationship is used to calculate the time required for the current (or voltage) to decrease by a certain percentage, such as the half-life of the current.

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