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An inductor-resistor (LR) series circuit consists of an inductor, with inductance denoted by 'L', and a resistor, with resistance 'R', connected one after the other. The behavior of this circuit becomes especially interesting when the circuit is switched on or off. When power is applied, the inductor opposes the change in current through it due to a property called 'inductance'. The opposition to change in current flow leads to a time-dependent change in current, which follows a mathematical model.

Understanding the basic property of inductors, which is to store energy in a magnetic field, is crucial to grasping how an LR series circuit operates. This stored energy affects how quickly the current can establish itself to the full, or final, current level, which is dependent upon the values of 'L' and 'R'. A higher inductance or lower resistance means a longer time for the current to rise, often described in terms of a time constant.

The current in an LR circuit doesn't immediately reach its maximum value. Instead, it gradually increases over time in a predictable fashion, known as an exponential approach to the final or steady-state current. For an inductor connected to a DC power source in series with a resistor, the current 'I(t)' at any given time 't' can be described by the formula: \( I(t) = I_0 \left(1 - e^{-\frac{t}{\tau}}\right) \), where \( I_0 \) is the final, steady-state current, \( e \) is the base of natural logarithms, and \( \tau \) is the LR time constant. The time constant \( \tau \) can be thought of as the 'speed' at which the current is changing. A smaller time constant means the current reaches its steady state more quickly. As a visual metaphor, if current change were a car, the time constant would be akin to its acceleration—determining how quickly it reaches cruising speed from a standstill.

In an LR series circuit, when the power supply is disconnected, the current doesn't drop to zero instantly. It diminishes in a process known as 'exponential decay'. This decay process is mirrored when the circuit is started, only in the reverse—it exponentially approaches the maximum current value. Mathematically, this behavior is captured by the 'decay' term \( e^{-\frac{t}{\tau}} \), which represents the proportion of current that has yet to transfer at time 't'.

After three time constants (\( t = 3\tau \) ), almost all (about 95%) of the current has established itself in the circuit. The exponential term \( e^{-3} \) is approximately 0.0498, meaning that 4.98% of the final current is what's left to reach the full current level. This exponential behavior is fundamental to understanding the transient response of inductors and overall LR circuit dynamics, enabling precise prediction and control of circuit behavior over time.