In the circuit shown in Fig. 32-37, switch S is closed at...

In the circuit shown in Fig. $32-37$, switch $S$ is closed at time $t=0$. Thereafter, the constant current source, by varying its emf, maintains a constant current $i$ out of its upper terminal. (a) Derive an expression for the current through the inductor as a function of time. (b) Show that the current through the resistor equals the current through the inductor at time $t=(L / R) \ln 2 .$

Key Concepts

Inductor Behavior and its Voltage-Current Relationship

An inductor resists changes in current due to its stored magnetic energy, a behavior mathematically expressed as V = L(di/dt). This property means that when a change in the applied circuit conditions occurs, the inductor causes the current to change gradually rather than instantaneously, which is critical in understanding the transient dynamics of the circuit.

Time Constant in RL Circuits

The time constant, defined as ? = L/R, is a key parameter that determines the rate at which currents in an RL circuit reach their steady state. It provides a measure of how quickly the transient effects decay, with a larger time constant indicating a slower transition.

RL Circuit Transient Response

This concept involves analyzing how the current and voltage in an RL circuit change over time immediately after a switching event. It focuses on how the circuit transitions from its initial state to a new steady state, with the behavior typically governed by exponential functions that describe the rise or decay of current.

First-Order Linear Differential Equations in Circuit Analysis

RL circuits are modeled by first-order linear differential equations derived from applying Kirchhoff’s voltage law. These equations involve terms that represent the inductor’s opposition to changes in current and the resistor’s voltage drop, and they are solved using methods such as the integrating factor, leading to solutions that typically involve exponential functions.

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