In the circuit shown in Fig. 30.11 switch S1 has been closed...

In the circuit shown in Fig. 30.11 switch $S_{1}$ has been closed a long time while switch $S_{2}$ has been left open. Then $S_{2}$ is closed at the same instant when $S_{1}$ is opened. Just after $S_{2}$ is closed, the current through the resistor is $12.0 \mathrm{~A}$ and its rate of decrease is $d i / d t=-36.0 \mathrm{~A} / \mathrm{s}$. How long does it take the current to decrease to $6.00 \mathrm{~A},$ one-half its initial value?

Key Concepts

RL Circuit Transient Response

This concept involves analyzing how the current and voltage in a resistor-inductor circuit change over time when a sudden change occurs, such as switching actions. The transient response describes the period during which the circuit variables move from one steady state to another, governed by the circuit’s inherent properties such as inductance and resistance.

Exponential Decay

In many RL circuits, when the circuit is subjected to a sudden change, the current does not drop instantly but rather decays gradually, following an exponential pattern. This behavior is described by an exponential function, where the rate of decrease in current is initially high and then gradually slows down as the current approaches its new steady state.

Time Constant in RL Circuits

The time constant, typically denoted by ?= L/R in an RL circuit, is a crucial parameter that determines the rate at which transient effects decay. It represents the characteristic time over which the current decays to approximately 36.8% of its initial value, providing a measure of how quickly the circuit responds to changes.

Differential Equations in Circuit Analysis

The behavior of transient circuits is mathematically modeled using differential equations. In an RL circuit, applying Kirchhoff’s voltage law leads to a first-order linear differential equation, which, upon solving, yields an exponential decay or growth formula characterizing the change in current over time. This approach is fundamental in predicting circuit behavior during switching operations.

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