91Ó°ÊÓ

In the realm of electric circuits, Ohm's Law is a fundamental principle that describes the relationship between voltage, current, and resistance. The law is mathematically expressed as \( V = I \times R \), where \( V \) is the voltage across the circuit, \( I \) is the current flowing through it, and \( R \) is the resistance present in the circuit.
One helpful way to understand Ohm's Law is to think of it as a bridge.
It connects the potential difference to the flow of electric charge, giving us a way to calculate the current if the voltage and resistance are known.
In our problem, before the battery is removed, the circuit is in a steady state with a constant current. Using Ohm’s Law, we calculate this initial current \( I_0 \) by dividing the battery voltage \( 12\, \text{V} \) by the circuit's total resistance \( R = 2200\, \Omega \).
- **Key Insight**: Ohm’s Law is crucial for determining how much current is flowing based on known voltage and resistance values. It provides the initial condition for analyzing further circuit dynamics.

Inductors play a critical role in circuits, storing energy in a magnetic field when current flows through them. When the current changes, an Electromotive Force (EMF) is induced in the inductor according to Lenz's Law.
This induced EMF opposes the change in current, described by the equation \( \mathscr{E} = -L \frac{dI}{dt} \), where \( L \) is the inductance and \( \frac{dI}{dt} \) represents the rate of change of current.
As the battery is removed from the circuit, the current begins to decrease, and consequently, the inductor produces an EMF to oppose this change. In our solution, \( \mathscr{E}(t) \) is directly derived from the decaying current \( I(t) \). By substituting the current decay expression into the EMF formula, we calculate how this EMF changes over time.
- **Key Insight**: Understanding how EMF works in inductors helps us see how inductors resist changes in current and how energy is dissipated when a battery is removed.

When a battery is disconnected from an LR circuit, the current doesn't stop immediately.
Instead, it decays exponentially over time, a pattern described by \( I(t) = I_0 e^{-\frac{t}{\tau}} \).
Here, \( I_0 \) is the initial current and \( \tau \) represents the time constant.
This exponential behavior is due to the energy stored in the inductor's magnetic field that continues to drive current even after the power source is removed.
The rate at which current decreases is determined by the circuit’s time constant \( \tau \).
The solution emphasizes how current follows this mathematical model, which is exceptionally common in circuits with inductors or capacitors.
Understanding this pattern is key to predicting how circuit quantities will behave over time, especially in circuits experiencing sudden component changes such as a power source disconnection.
- **Key Insight**: Recognizing exponential decay in current allows us to predict circuit behavior accurately after changes, using foundational mathematical relationships.

The time constant, \( \tau \), is an essential parameter in the analysis of LR circuits. It is calculated as \( \tau = \frac{L}{R} \), where \( L \) is the inductance and \( R \) is the resistance.
The time constant defines the speed at which the current decreases in the circuit. In simpler terms, it represents the time it takes for the current to fall to around 37% of its initial value \( I_0 \).
In our given problem, the calculated time constant is \( 8.18\, \mu s \).
This explains how quickly the circuit loses current after the battery is removed.
Knowing \( \tau \) aids in understanding how fast responses occur and how long one can expect energy storage in inductors.
This insight is crucial for designing circuits with specific timing requirements.
- **Key Insight**: The time constant is a direct measure of the pace at which reactive elements (like inductors) respond to changes.
Grasping \( \tau \) is vital for predicting how quickly circuit dynamics settle after disturbances.