If, in Fig. 34-2, R=20 Ω, L=0.30 H, and ε=90 V, what will be the current in the circuit 0.050

step 1 Hi, in this given problem here, the value of resistance is 20 oom inductance, that is 0 .30 Henr

If, in Fig. 34-2, R=20 Ω, L=0.30 H, and ε=90 V, what will...

If, in Fig. 34-2, $R=20 \Omega$, $L=0.30 \mathrm{H}$, and $\varepsilon=90 \mathrm{~V}$, what will be the current in the circuit $0.050 \mathrm{~s}$ after the switch is closed? We are going to use the exponential equation for $i$ given in Eq. (34.9). The time constant for this circuit is $\mathrm{L} / \mathrm{R}=0.015 \mathrm{~s}$, and $i_{\infty}=\varepsilon / \mathrm{R}=$ $4.5 \mathrm{~A}$. Then $$ i=i_{x}\left(1-e^{-i /\langle L / R)}\right)=(4.5 \mathrm{~A})\left(1-e^{-333}\right)=(4.5 \mathrm{~A})(1-0.0357)=4.3 \mathrm{~A} $$

If, in Fig. 34-2, $R=20 \Omega$, $L=0.30 \mathrm{H}$, and $\varepsilon=90 \mathrm{~V}$, what will be the current in the circuit $0.050 \mathrm{~s}$ after the switch is closed?
We are going to use the exponential equation for $i$ given in Eq. (34.9).
The time constant for this circuit is $\mathrm{L} / \mathrm{R}=0.015 \mathrm{~s}$, and $i_{\infty}=\varepsilon / \mathrm{R}=$
$4.5 \mathrm{~A}$. Then
$$
i=i_{x}\left(1-e^{-i /\langle L / R)}\right)=(4.5 \mathrm{~A})\left(1-e^{-333}\right)=(4.5 \mathrm{~A})(1-0.0357)=4.3 \mathrm{~A}
$$

Key Concepts

RL Circuit Analysis

This concept involves examining circuits that include resistors and inductors. Key aspects include how these components interact, particularly when a voltage source is connected, influencing both the transient and steady-state responses of the circuit. RL circuits are often analyzed with differential equations to understand how current changes over time as energy is stored in the magnetic field of the inductor.

Time Constant

The time constant in an RL circuit is defined as the ratio of the inductance (L) to the resistance (R). It characterizes how quickly the current reaches its steady-state value. A smaller time constant means the circuit responds more quickly, while a larger time constant indicates a slower response, as it represents the rate at which the transient effects decay exponentially with time.

Exponential Response in RL Circuits

When an RL circuit is energized, the current does not instantaneously reach its final value but increases exponentially. This behavior is described by an equation of the form i = I_infinity*(1 - exp(-t/(L/R))), where I_infinity is the steady-state current. This exponential model captures how the inductor resists abrupt changes in current, leading to a gradual buildup over time.

Steady-State Current

Steady-state current is the current that flows through an RL circuit after the transient effects have died out, typically calculated using Ohm's law (I = V/R) when the inductor no longer opposes the current change. In this state, the circuit behaves as if it were purely resistive, and the inductor acts like a short circuit with no voltage drop.

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