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Exponential decay refers to the process where the current in an LR circuit decreases over time after reaching a transient state. In an LR (inductor-resistor) circuit, this decay occurs as the circuit responds to changes in voltage after the switch is closed. The behavior of this current decay is modeled by the equation:
\[ i(t) = i_{\infty} \left(1 - e^{-t / \tau}\right)\]Here, \(i_{\infty}\) is the steady-state current, and \(\tau\) is the time constant. The term \(e^{-t/\tau}\) represents the exponential decay part.

• Exponential decay is characterized by a rapid drop in current initially, slowing over time.
• It is a gradual process that's asymptotic, meaning the current approaches a value indefinitely without reaching it.
• This concept is crucial for understanding how inductors resist changes in current flow, affecting the overall energy dynamics inside the circuit.

Understanding exponential decay is essential for circuit analysis as it helps predict how quickly a circuit component like an inductor or resistor stabilizes to a new condition.

The time constant, often denoted by \(\tau\), is a pivotal parameter in analyzing an LR circuit. It effectively dictates the rate at which current changes occur and how swiftly a circuit reaches its steady state after the switch is closed.

• The time constant is calculated as the ratio of inductance \(L\) to resistance \(R\):\[ \tau = \frac{L}{R}\]
• It serves as a measure of how fast or slow the exponential decay of current happens in the circuit.
• In our example, the time constant was calculated as \(0.015\, \text{s}\).

If the time constant is small, the circuit reaches its steady-state rapidly, indicating a fast response. A larger time constant means a slower response time.Due to its importance:

• A time constant of \(\tau\) seconds means that after this period, the current will have completed about 63.2% of its decay towards the final value.
• In practical applications, after approximately 5 time constants, the transient effects are considered negligible and the circuit is said to have reached its steady state.

Understanding the time constant is crucial for predicting the behavior of electrical circuits in a variety of engineering and physics applications.

The steady-state current, denoted as \(i_{\infty}\), is the current that flows through the circuit once all transient behaviors have subsided, and the circuit is in full equilibrium.
In our LR circuit, the steady-state current can be determined by Ohm’s Law once the inductive effects are negligible:

• The formula used is:\[i_{\infty} = \frac{\mathscr{E}}{R}\]
• Where \(\mathscr{E}\) is the electromotive force (emf) or the battery voltage and \(R\) is the resistance.

For our problem:

• The given values result in a steady-state current of \(4.5\, \text{A}\).
• It is the maximum current that would flow through the circuit if left undisturbed indefinitely.

The concept of steady-state current is fundamental to appreciate circuit behavior over a longer time frame. It's especially useful for scenarios where analysis of the final, unchanging current is required, such as in power calculations and ensuring component ratings are appropriate for continuous operation.