Consider an R L circuit (Eig. P 21.38 ) with V=9.0 V, L=5.5...

Consider an $R L$ circuit (Eig. $\mathrm{P} 21.38$ ) with $V=9.0 \mathrm{V}, L=5.5 \mathrm{mH},$ and $R=1500 \Omega$. After the switch has been left open for a long time, the switch is closed at $t=0$ a. What is the current the instant after the switch is closed? b. What is the current a very long time after the switch is closed? c. How long does it take the current in the circuit to reach $50 \%$ of its final (long time) value?

Consider an $R L$ circuit (Eig. $\mathrm{P} 21.38$ ) with $V=9.0 \mathrm{V}, L=5.5 \mathrm{mH},$ and $R=1500 \Omega$. After the switch has been left open for a long time, the switch is closed at $t=0$
a. What is the current the instant after the switch is closed?
b. What is the current a very long time after the switch is closed?
c. How long does it take the current in the circuit to reach $50 \%$ of its final (long time) value?

Key Concepts

Time Constant

The time constant in an RL circuit, defined as tau (?) = L/R, represents the time it takes for the current to reach approximately 63.2% of its final steady-state value after a change is introduced. It is a key parameter in analyzing the exponential approach of the current to its final value, and it quantifies the speed of the transient response.

Steady-State Behavior

For an RL circuit subjected to a constant applied voltage, the steady-state behavior is achieved once the effects of the inductor have diminished over time. In the steady state, the inductor behaves like a short circuit, meaning that the final current is determined solely by Ohm's law (I = V/R). This concept is critical in understanding how the circuit settles after a transient phase.

RL Circuit

An RL circuit is an electrical circuit composed of a resistor (R) and an inductor (L) connected in series or parallel. These circuits are often analyzed to study transient phenomena, where the inductor initially opposes changes in current. The behavior of an RL circuit is typically characterized by the interplay between the resistive and inductive elements, and it is governed by differential equations that describe how current changes over time.

Transient Response

The transient response of an RL circuit describes how the current evolves from an initial state to a steady state after a sudden change, such as closing a switch. In these circuits, the current does not change instantaneously due to the inductor’s opposition to sudden changes in current. The transient behavior is usually characterized by an exponential function that depends on the circuit's time constant.

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