*Notes to a video lecture on http://www.unizor.com*

__Alternating Current and Inductors__

For the purpose of this lecture it's important to be familiar with the concept of a

*self-induction*explained in "Electromagnetism - Self-Induction" chapter of this course.

In this lecture we will discuss the AC circuit that contains an

*inductor*- a wire wound in a reel or a solenoid, thus making multiple loops, schematically presented on the following picture.

Both direct and alternating current go through an inductor, but, while

direct current goes with very little resistance through a wire, whether

it's in a shape of a loop or not, alternating current meets some

additional resistance when this wire is wound into a loop.

Consider the following experiment.

Here the AC circuit includes a lamp, an inductor in a shape of a solenoid and an iron rod fit to be inserted into a solenoid.

While the rod is not inside a solenoid, the lamp lights with normal

intensity. But let's gradually insert an iron core into a solenoid. As

the core goes deeper into a solenoid, the lamp produces less and less

light, as if some kind of resistance is increasing in the circuit.

This experiment demonstrates that inductors in the AC circuit produce

effect similar to resistors, and the more "inductive" the inductor - the

more resistance can be observed in a circuit.

The theory behind this is explained in this lecture.

The cause of this resistance is

*self-induction*. This concept was

explained earlier in this course and its essence is that variable

magnetic field flux, going through a wire loop, creates electromotive

force (EMF) directed against the original EMF that drives electric

current through a loop.

Any current that goes along a wire creates a magnetic field around this

wire. Since the current in our wire loop is alternating, the magnetic

field that goes through this loop is variable. According to the

Faraday's Law, the variable magnetic field going through a wire loop

generates EMF equal to a rate of change of the magnetic field flux and

directed opposite to the EMF that drives the current through a wire,

thus resisting it.

Magnetic flux

*going through inductor, as a function of time*

**Φ(t)***, is proportional to an electric current*

**t***going through its wire*

**I(t)**

**Φ(t) = L·I(t)**where

*is a coefficient of proportionality that depends*

**L**on physical properties of the inductor (number of loop in a reel, type

of its core etc.) called

*inductance*of the inductor.

If the current is alternating as

**I(t) = I**_{max}·sin(ωt)the flux will be

**Φ(t) = L·I**_{max}·sin(ωt)According to Faraday's Law, self-induction EMF

**E**_{i}is equal in magnitude to a rate of change of magnetic flux and opposite

in sign (see chapter "Electromagnetism - "Self-Induction" in this

course)

**E**d_{i}(t) = −**Φ/**d**t =**

= −L·d= −L·

**I(t)/**d**t =**

= −L·ω·I

= −L·ω·I

= −E= −L·ω·I

_{max}·cos(ωt) == −L·ω·I

_{max}·sin(ωt+π/2) == −E

_{imax}·sin(ωt+π/2)where

**E**_{imax}= L·ω·I_{max}The unit of measurement of

*inductance*is

**henry (H)**with

**1H**being an inductance of an inductor that generates

**1V**electromotive force, if the rate of change of current is

**1A/sec**.

That is,

*henry = volt·sec/ampere = ohm·sec*

An expression

*in the above formula for*

**X**_{L}=L·ω*is called*

**E**_{i}*inductive reactance*. It plays the same role for an inductor as

*resistance*for resistors.

The units of the inductive reactance is

**Ohm (Ω)**because

*henry/sec = ohm·sec/sec = ohm*.

Using this concept of

*inductive reactance*

*of an*

**X**_{L}*inductor*, the time dependent induced EMF is

**E**_{i}(t) = −X_{L}·I_{max}·sin(ωt+π/2) = −E_{imax}·sin(ωt+π/2)and

*,*

**E**_{imax}= X_{L}·I_{max}which for

*inductors*in AC circuit is an analogue of the Ohm's Law for

*resistors*.

What's most important in the formula

**E**_{i}(t) = −E_{imax}·sin(ωt+π/2)and the most important property of an inductor in an AC circuit is

that, while the electric current in a circuit oscillates with angular

speed

*, the*

**ω****voltage drop on an inductor oscillates with the same angular speed**.

*as the current, but its period is shifted in time by***ω***π/2*relative to the current
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