Consider the R L circuit of Fig. 32-6. In terms of the battery emf ℰ, (a) what is the self-induc

step 1 Consider the RL circuit which is in the form there is the switch given followed by the resistanc

Consider the R L circuit of Fig. 32-6. In terms of the...

Consider the $R L$ circuit of Fig. 32-6. In terms of the battery emf $\mathscr{E}$, (a) what is the self-induced emf $\Delta V_{2}$ when the switch has just been closed on $a$, and (b) what is $\Delta V_{2}$ when $t=2.0 \tau_{L} ?$ (c) In terms of $\tau_{L}$, when will $\Delta V_{2}$ be just one-half the battery emf $\mathscr{E}$ ?

Key Concepts

RL Circuit

An RL circuit is an electrical circuit consisting of a resistor (R) and an inductor (L) connected in series or parallel. These circuits are used to study the behavior of current and voltage when subjected to changes, such as when a switch is closed, and they exhibit characteristic transient responses due to the interplay between resistance and inductance.

Time Constant in RL Circuits

The time constant (?) in an RL circuit is defined as ? = L/R, where L is the inductance and R is the resistance. This time constant characterizes the rate at which currents and voltages in the circuit approach their steady state; it represents the time needed for the system’s response to reach about 63.2% of its final value during a transient process.

Self-Induced EMF

Self-induced electromotive force (EMF) in an inductor arises due to a change in the current passing through it. According to Faraday’s law of electromagnetic induction, this induced voltage opposes the change in current, a phenomenon known as Lenz’s law. This behavior is crucial in understanding the transient responses in inductive circuits.

Transient Response

Transient response refers to the short-term behavior of a circuit immediately after a change in its conditions, such as closing a switch. In an RL circuit, the current (and the associated voltage across the inductor) does not change instantaneously but rather follows an exponential curve determined by the circuit’s time constant. This response provides insight into how quickly the circuit reaches its steady state after a disturbance.

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