A 24 V battery is connected in series with a resistor and an...

A 24 V battery is connected in series with a resistor and an inductor, with $R=8.0 \Omega$ and $L=4.0 \mathrm{H}$ , respectively. Find the energy stored in the inductor (a) when the current reaches its maximum value and (b) one time constant after the switch is closed.

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Key Concepts

Inductive Energy Storage

This concept refers to the energy that is stored in the magnetic field of an inductor. The energy is quantified by the formula E = 1/2 L I², where L represents the inductance and I represents the current flowing through the inductor. It is fundamental in understanding how energy is accumulated and later released in inductive components.

Time Constant of an RL Circuit

The time constant, denoted by ? and defined by ? = L/R, characterizes how quickly current builds up or decays in a resistor-inductor circuit following a sudden change, such as closing a switch. It indicates the time it takes for the current to reach about 63% of its final value, highlighting the rate of the transient response in RL circuits.

Transient and Steady-State Analysis

This concept involves differentiating between the transient phase—immediately after a circuit change where current varies exponentially—and the steady state, where the current becomes constant. In an RL circuit, the transient phase is governed by the time constant, while in steady state the inductor behaves like a simple conductor with negligible resistance, setting the maximum current determined by the resistor and applied voltage.

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