Equal of self inductance l is connected at the one end of two real as shown in the figure

When discussing self-inductance and its connection in circuits, it’s essential to understand how inductors behave in different configurations. In your scenario, you mentioned that an inductor with self-inductance \( L \) is connected at one end to two resistors. Let’s break this down step by step to clarify the concepts involved.

Understanding Self-Inductance

Self-inductance is a property of an inductor that quantifies its ability to store energy in a magnetic field when an electric current flows through it. The self-inductance \( L \) is measured in henries (H) and indicates how much voltage is induced in the inductor for a given rate of change of current.

Configuration of the Circuit

In your case, if the inductor is connected at one end to two resistors, we need to consider how these components interact. Let’s assume the other end of the inductor is connected to a voltage source. The configuration can significantly affect the overall behavior of the circuit.

• Series Connection: If the resistors are in series with the inductor, the total resistance \( R_{total} \) is simply the sum of the resistances of the two resistors. The total inductance in this case remains \( L \), and the circuit will respond according to the combined resistance and inductance.
• Parallel Connection: If the resistors are connected in parallel with each other and the inductor, the equivalent resistance \( R_{eq} \) can be calculated using the formula \( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \). The inductor still has its self-inductance \( L \), but the overall circuit dynamics will change due to the reduced equivalent resistance.

Analyzing the Circuit Behavior

Let’s consider the implications of these configurations:

• Transient Response: When the circuit is first energized, the inductor will resist changes in current due to its inductance. The time constant \( \tau \) for the circuit can be calculated as \( \tau = \frac{L}{R_{total}} \) for a series connection or \( \tau = \frac{L}{R_{eq}} \) for a parallel connection. This time constant indicates how quickly the current will reach its steady-state value.
• Steady-State Behavior: After a long time, the inductor will behave like a short circuit (assuming a DC source), and the current through the circuit will be determined by Ohm’s law, \( I = \frac{V}{R_{total}} \) or \( I = \frac{V}{R_{eq}} \), depending on the configuration.

Practical Example

Imagine you have an inductor of 2 H connected to two resistors, one of 4 ohms and the other of 6 ohms. If these resistors are in series with the inductor, the total resistance is \( 4 + 6 = 10 \) ohms. The time constant would be \( \tau = \frac{2}{10} = 0.2 \) seconds. This means it will take about 0.2 seconds for the current to rise significantly towards its maximum value.

On the other hand, if the resistors are in parallel, the equivalent resistance would be \( R_{eq} = \frac{1}{\frac{1}{4} + \frac{1}{6}} = 2.4 \) ohms. The time constant in this case would be \( \tau = \frac{2}{2.4} \approx 0.83 \) seconds, indicating a different rate of current change.

Conclusion

Understanding how self-inductance interacts with resistors in different configurations is crucial for analyzing circuits. By applying the principles of inductance and resistance, you can predict how the circuit will behave over time, both during transient and steady-state conditions. This knowledge is foundational for further studies in electrical engineering and physics.