In the Circuit In the circuit of Fig. 32-33, ℰ=10 V, R1=5.0 Ω, R2 =10 Ω, and L=5.0 H. For the t

step 1 In the given circuit, the switch is given and here are the two branches in which one is the resi

In the Circuit In the circuit of Fig. 32-33, ℰ=10 V, R1=5.0...

In the Circuit In the circuit of Fig. $32-33, \mathscr{E}=10 \mathrm{~V}, R_{1}=5.0 \Omega, R_{2}$ $=10 \Omega$, and $L=5.0 \mathrm{H}$. For the two separate conditions (I) switch S just closed and (II) switch S closed for a long time, calculate (a) the current $i_{1}$ through $R_{1}$, (b) the current $i_{2}$ through $R_{2},(\mathrm{c})$ the current $i$ through the switch, (d) the potential difference across $R_{2}$, (e) the potential difference across $L$, and $(\mathrm{f})$ the rate of change $d i_{2} / d t$.

In the Circuit In the circuit of Fig. $32-33, \mathscr{E}=10 \mathrm{~V}, R_{1}=5.0 \Omega, R_{2}$
$=10 \Omega$, and $L=5.0 \mathrm{H}$. For the two separate conditions (I) switch S just closed and (II) switch S closed for a long time, calculate (a) the current $i_{1}$ through $R_{1}$, (b) the current $i_{2}$ through $R_{2},(\mathrm{c})$ the current $i$ through the switch, (d) the potential difference across $R_{2}$, (e) the potential difference across $L$, and $(\mathrm{f})$ the rate of change $d i_{2} / d t$.

Key Concepts

Transient Response in RL Circuits

When the switch in an RL circuit is first closed, the inductor initially opposes changes in current due to its stored magnetic energy. This results in a transient period where the current builds up gradually rather than instantaneously reaching its final value. The rate at which the current changes is governed by the inductor's characteristics and the circuit’s resistance, influencing the initial behavior immediately after closing the circuit.

Steady-State Analysis of RL Circuits

After the circuit has been closed for a long time, the transient effects vanish, and the circuit reaches a steady state where the current is constant. In this condition, the inductor behaves like a short circuit (i.e., it offers negligible impedance to constant current). This simplifies the analysis, as the voltage drops across the resistive components can be determined directly from Ohm’s law without accounting for the inductive effects.

Kirchhoff’s Circuit Laws

Kirchhoff’s Voltage Law (KVL) and Kirchhoff’s Current Law (KCL) are foundational for analyzing circuits, including those involving inductors, resistors, and voltage sources. KVL is used to sum the potential differences around a closed loop, ensuring that they equal zero, while KCL helps in distributing the current correctly at circuit junctions. These laws are critical in determining the current and voltage distributions in both transient and steady-state conditions.

Time Constant in RL Circuits

The time constant of an RL circuit, defined as the ratio of the inductor’s inductance (L) to the total resistance (R) in the circuit, characterizes the speed at which the circuit approaches its steady state. A larger time constant means that the transient period will be longer, while a smaller time constant allows the circuit to reach its final steady-state current more quickly.

Inductor Voltage and Back EMF

The voltage across an inductor is directly related to the rate of change of its current, described by the equation V = L (di/dt). This relationship underpins the concept of back electromotive force (back EMF), which is the inductor’s inherent resistance to changes in current. It plays a crucial role in the transient behavior of the circuit, influencing how rapidly or slowly the current evolves after the switch is closed.

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