Once the switch S is closed in Fig. 32-36 the time required...

Once the switch $\mathrm{S}$ is closed in Fig. $32-36$ the time required for the current to reach any obtainable value depends, in part, on the value of resistance $R$. Suppose the emf $\mathscr{E}$ of the ideal battery is $12 \mathrm{~V}$ and the inductance of the ideal (resistanceless) inductor is $18 \mathrm{mH}$. FIGURE $32-36=$ How much time is needed for the current Problems 31,34 , to reach $2.00 \mathrm{~A}$ if $R$ is (a) $1.00 \Omega$, (b) $5.00$ and 61 . $\Omega$, and $(\mathrm{c}) 6.00 \Omega ?(\mathrm{~d}) \mathrm{Why}$ is there a huge jump between the answers to (b) and (c)? (e) For what value of $R$ is the time required for the current to reach $2.00 \mathrm{~A}$ least? (f) What is that least time? (Hint: Rethink Eq. 32-21.)

Once the switch $\mathrm{S}$ is closed in Fig. $32-36$ the time required for the current to reach any obtainable value depends, in part, on the value of resistance $R$. Suppose the emf $\mathscr{E}$ of the ideal battery is $12 \mathrm{~V}$ and the inductance of the ideal (resistanceless) inductor is $18 \mathrm{mH}$. FIGURE $32-36=$ How much time is needed for the current Problems 31,34 , to reach $2.00 \mathrm{~A}$ if $R$ is (a) $1.00 \Omega$, (b) $5.00$ and 61 . $\Omega$, and $(\mathrm{c}) 6.00 \Omega ?(\mathrm{~d}) \mathrm{Why}$ is there a huge
jump between the answers to (b) and (c)? (e) For what value of $R$ is the time required for the current to reach $2.00 \mathrm{~A}$ least? (f) What is that least time? (Hint: Rethink Eq. 32-21.)

Key Concepts

RL Circuit Behavior

An RL circuit involves a resistor and an inductor connected in series with a voltage source. When the circuit is closed, the inductor initially opposes changes in current, and the resulting behavior is governed by a first?order differential equation that describes how the current builds up over time until reaching a steady state.

Time Constant

The time constant, denoted by ? and defined as L/R (inductance divided by resistance), characterizes the rate at which the current in an RL circuit approaches its final value. It serves as a measure of the system's response speed; a larger time constant means a slower current rise, while a smaller one implies a quicker transition.

Exponential Transient Response

The current in an RL circuit increases according to an exponential function, ultimately approaching its steady-state value asymptotically. After one time constant, the current typically reaches about 63% of its final value, and this exponential nature is fundamental to understanding the transient (non?steady state) behavior in such circuits.

Parameter Sensitivity in RL Circuits

The value of the resistance R in an RL circuit not only determines the final steady-state current but also has a significant impact on the time constant. Small variations in resistance can lead to disproportionately large changes in the rate at which current builds up, leading to 'huge jumps' in the time required to reach a specified current level when R is varied.

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