In an inductive coil, the rate of change of current is maximum:

Explanation:

Inductive Coil and Rate of Change of Current

Definition: An inductive coil, often simply referred to as an inductor, is a passive electrical component that stores energy in its magnetic field when electrical current passes through it. It resists changes in the current flowing through it due to its property known as inductance.

Working Principle: When a voltage is applied across an inductor, it creates a time-varying magnetic field, resulting in an induced electromotive force (emf) that opposes the change in current. This phenomenon is described by Faraday's Law of Electromagnetic Induction and Lenz's Law.

Correct Option Analysis:

The correct option is:

Option 2: At the start of current flow.

This option accurately describes the behavior of an inductive coil when a voltage is first applied. At the initial moment of current flow, the rate of change of current is at its maximum. This can be understood by examining the fundamental principles of inductance and the behavior of an RL (resistor-inductor) circuit.

When a voltage \( V \) is applied to an inductive coil with inductance \( L \) and resistance \( R \), the current \( I(t) \) through the coil at any time \( t \) is given by the differential equation:

\( V = L \frac{dI(t)}{dt} + IR \)

At the moment the voltage is applied (\( t = 0 \)), the current \( I(0) \) is zero, and the rate of change of current \( \frac{dI(0)}{dt} \) is at its peak because the entire voltage is initially dropped across the inductor:

\( V = L \frac{dI(0)}{dt} \)

Thus, the maximum rate of change of current occurs at \( t = 0 \), immediately after the voltage is applied.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: After one time constant.

This option is incorrect. The time constant (\( \tau = \frac{L}{R} \)) of an RL circuit represents the time required for the current to reach approximately 63.2% of its maximum value. After one time constant, the rate of change of current is not at its maximum; it has significantly decreased as the current approaches its steady-state value.

Option 3: Near the final maximum value of current.

This option is also incorrect. As the current approaches its final steady-state value, the rate of change of current diminishes and approaches zero. At the maximum current value, the inductor behaves almost like a short circuit, and there is no further change in current.

Option 4: At 36.8% of its maximum steady-state value.

This option is incorrect. At 36.8% of its maximum steady-state value (which corresponds to \( 1 - \frac{1}{e} \)), the rate of change of current is not at its maximum. The maximum rate of change occurs at the very start, not at this particular value of current.

Conclusion:

Understanding the behavior of inductive coils in electrical circuits is essential for analyzing how current changes over time. The rate of change of current in an inductor is highest at the start of current flow, immediately after a voltage is applied. This behavior is crucial for the design and analysis of circuits involving inductive components, including those used in power supplies, transformers, and various types of electronic equipment.