In Fig. 27.26, take ℰ0=11 V, R=3.0 Ω, and L=18 H. Initially the switch is in position B and the

step 1 Hey everyone, so here we have the time constant. So we have t is equals to 1 divided by r. So we

In Fig. 27.26, take ℰ0=11 V, R=3.0 Ω, and L=18 H. Initially...

In Fig. $27.26$, take $\mathcal{E}_{0}=11 \mathrm{~V}, R=3.0 \Omega$, and $L=18 \mathrm{H}$. Initially the switch is in position $B$ and there's no current anywhere. At $t=0$ the switch is thrown to position $A$, and, at $t=15 \mathrm{~s}$, it's returned to $B$. Find the inductor current at (a) $t=7.50 \mathrm{~s}$ and (b) $t=22.5 \mathrm{~s}$.

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Key Concepts

Exponential Behavior of Current

In RL circuits, the current evolves following an exponential function. When a voltage is applied, the current rises as I(t) = I_final (1 ? e^(?t/?)), whereas when the voltage is removed or the circuit configuration changes, the current decays exponentially. This behavior is essential for predicting the current at any given time following a switching event.

RL Circuit Transient Response

This concept involves understanding how current changes over time in a resistor?inductor (RL) circuit when subject to a sudden change, such as closing a switch. The transient response is governed by a differential equation that describes how the current builds up or decays exponentially to its steady-state value after the circuit configuration is altered.

Time Constant (?) in RL Circuits

The time constant, defined as ? = L/R, is crucial for quantifying the speed of the transient process in an RL circuit. It characterizes the time it takes for the current to reach approximately 63% of its final value during the growth phase (or decay to about 37% during the decay phase), indicating how quickly the circuit responds to switching events.

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