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The coil being an inductive load has an "inrush" of current when first energized that rapidly decreases as the magnetic field increases.
This is not a correct statement.
One characteristic of an inductor is that its current can not be instantaneously changed.
Thus, in a simple RL series circuit with an open switch, battery, and initial current of zero there is no energy stored in the inductor and the voltage across the inductor is zero.
The instant after the switch closes there is no current flow. The boundary conditions are identical on both sides of t=0. The next instant the current starts to rise. The current rise is exponential with time until at infinite time the steady state current is the battery voltage divided by the series resistance.
At one time constant the current will have risen to about 63% of its steady state value. During the next time constant period the current rises an additional 63% of the remaining current change. After 6 time constants the current is within 0.25% of its steady state value. For a simple series RL series circuit the time constant is L/R.
In contrast for a capacitor in a simple series RC circuit with zero initial charge on the capacitor the initial current after the switch closes is a maximum equal to the source voltage divided by the series resistance.
When a ferromagnetic inductive circuit is considered there are other factors to consider, but the basic characteristics of an inductive circuit are not violated.
Consider a transformer with an iron core, steel if you please. If the iron core is removed, then the air core inductor behaves as described above. When the core is installed, then the inductance is greatly increased. However, the inductance is a function of the flux density in the core. At a sufficiently high flux density the the coil looks like an air core inductor.
Suppose the transformer excitation is turned off such that there is a large residual flux in the core. Next apply a voltage to this coil of a polarity that forces the core further into saturation, then after a short time the input current will be much higher than if the voltage had been of the opposite polarity which would have caused a lowering of the flux density. This relates to the hysteresis curve of the magnetic material.
For a relay or solenoid it is somewhat different. Here there is a mechanical change that causes a change of inductance with time. When the relay or solenoid is de-energized there is a large air gap and the inductance is low. Thus, a high current after a short time. As the armature closes the inductance rises and the current diminishes.
An oscilloscope and some parts will allow a study of some of these characteristics.
You can learn more about these circuits in various books. One such book for circuit calculations is "Analysis of A-C Circuits", by Melville B. Stout, Ulrich's Book Store, 1952. One covering ferromagnetic characteristics is "Electric and Magnetic Fields", by Steven S. Attwood, John Wiley & Sons, 1949.
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