In Fig. 27.23 a, take R=2.5 k Ωand ℰ0=50 V. When the switch is closed, the current through the

step 1 Hi, here in this given problem the value of resistance in the circuit that is 2 .5 kilo ohm emf

In Fig. 27.23 a, take R=2.5 k Ωand ℰ0=50 V. When the switch...

In Fig. $27.23 a$, take $R=2.5 \mathrm{k} \Omega$ and $\mathcal{E}_{0}=50 \mathrm{~V}$. When the switch is closed, the current through the inductor rises to $10 \mathrm{~mA}$ in $30 \mu$. Find (a) the inductance and (b) the current in the circuit after many time constants.

Key Concepts

Steady-State Behavior

After the passage of several time constants, an RL circuit reaches its steady state, where the effects of inductance become negligible and the inductor behaves like a short circuit. In this state, the current is determined solely by the resistor and the applied voltage, calculated as I = E/R. The steady-state analysis simplifies circuit behavior by focusing on the long-term outcome after the transient effects have decayed.

Transient Response Analysis

Transient analysis in an RL circuit focuses on the behavior of the circuit immediately after a change in conditions, such as when a switch is closed. During the transient period, the current increases exponentially according to the formula i(t) = I_final*(1 - exp(-t/?)), where I_final is the steady-state current. This analysis is crucial for understanding how inductors resist sudden changes in current.

Time Constant in RL Circuits

The time constant for an RL circuit is defined as ? = L/R, where L is the inductance and R is the resistance. This constant characterizes the rate at which the current approaches its steady-state value following a sudden change, such as the closing of a switch. It represents the time it takes for the current to reach about 63% of its final steady-state value.

RL Circuits

An RL circuit consists of a resistor and an inductor connected in series or parallel and is commonly used to analyze circuits with time-varying currents. The inductor opposes changes in current by generating an electromotive force (emf), while the resistor limits the overall current in the circuit. Understanding RL circuits involves setting up and solving differential equations that describe the current and voltage dynamics over time.

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