A battery, switch, resistor, and inductor are connected in series. When the switch is closed, the current rises to half its steady state value in 1.0 ms. How long does it take for the magnetic energy in the inductor to rise to half its steady-state value?

When it comes to understanding the role of inductors in circuits, it's essential to grasp their core function: to store energy in a magnetic field when electric current passes through them. Picture an inductor as a kind of 'energy reservoir' that gradually fills up when a current is applied and can release this energy when the circuit conditions change.

Inductors oppose changes in current because of a property called 'inductance,' denoted by the symbol L. When a switch in a circuit containing an inductor is closed, the inductor does not immediately let the current reach its maximum value; instead, it increases over time, typically following an exponential curve. This unique behavior greatly affects the dynamics of electronic circuits, from filtering unwanted signals to managing power effectively in various devices.

The concept of steady-state current is central to the analysis of electrical circuits. This term refers to the current in a circuit component after all transient effects have died out. In other words, it's the constant current that flows through the component when it has been connected to a steady voltage source for a sufficiently long time and all oscillations or variances have settled.

Understanding steady-state current allows us to predict and model the long-term behavior of a circuit. It's the baseline from which any change or response can be measured. For example, in an inductor, as the steady-state is approached, the stored magnetic energy reaches its maximum, and the voltage across the inductor drops to zero, because at this point, the inductor behaves as a regular conducting wire with the entire source voltage dropping across other circuit elements.

The relationship between energy and current in inductors is a foundational concept in the field of electromagnetism and circuit analysis. The energy (W) stored in an inductor is directly proportional to the inductance (L) of the inductor and the square of the current (I) flowing through it, as expressed by the formula \( W = 0.5 L I^2 \).

This squared relationship indicates that even small increases in current lead to significant increases in the energy stored within the inductor's magnetic field. It is this quadratic dependency that causes the time needed for the energy to reach half of its steady-state value to differ from the time it takes for the current to reach half of its steady value. Notably, this discrepancy is well-illustrated by the given exercise, where it employs an understanding of exponential functions to arrive at the correct time duration required for the magnetic energy to reach half of its maximum potential.