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

__Self-Induction__

Let's study the process of increasing the direct electric current in a

wire loop that is connected to a source of a direct electricity (like

battery) that generates electromotive force of voltage

*.*

**U**_{0}This increase might be attributed to a decrease in resistance in this

loop. For example, if the wire loop was initially disconnected from a

source of electricity and we connected it, flipping some switch, the

resistance is changing from being practically infinite to some value

*during a fraction of a second and the electric current would grow from zero to some value*

**R***(Ohm's Law).*

**I = U/R**The first thing that we can observe is that there is a magnetic field

generated by an electric current with magnetic field lines going through

a loop, thus effectively making it a magnet.

The next thing we note is that the intensity of a magnetic field

generated by an increasing direct current is also increasing with the

current because of their relationship that we have described in the

lecture about magnetism of an electric current loop.

In the center of an ideally circular loop this intensity, as we calculated in that lecture, is

**B = μ**_{0}·I/(2R)where

*is*

**μ**_{0}*permeability*of space,

*is the current in a wire loop,*

**I***is the radius of a wire loop.*

**R**At other points around the wire the intensity of a magnetic field also

increases proportionally to the electric current in a wire. Even if the

shape of our wire loop is not ideally circular, the proportionality of

intensity of a magnetic field at any point to an electric current would

still be held, because each infinitesimal segment of a wire creates its

own magnetic field and the magnetic field intensity is an additive

function, that is, produced by two sources, intensities from each are

added as vectors, resulting in a combined intensity.

As a result, the

*magnetic flux*going through a wire loop increases proportionally to an increase in electric current.

Now recall the Faraday's Law of

**magnetic induction**. It states that changing

*magnetic flux*going through an electric circuit generates

*electromotive force (EMF)*proportional to a

**rate of change**of the

*magnetic flux*, where

**rate of change**means a change per unit of time, which, using mathematical language, is the first derivative of a variable

*magnetic flux*by time.

Let's ignore for now the signs of the rate of change of the

*magnetic flux*and the

*EMF*, then the Faraday's Law can be represented as

**U =**d**Φ/**d**t**where

*is the absolute value of a magnitude of the generated*

**U***EMF*,

*is*

**Φ***magnetic flux*,

*is time,*

**t***d*is the rate of change of magnetic flux (first derivative of magnetic flux by time), taken as an absolute positive value.

**Φ/**d**t**From

(1) the proportionality of a magnetic field intensity at any point around a wire to an electric current going through a wire,

(2) definition of magnetic flux and

(3) the Faraday's Law

follows that the increasing electric current, causing an increasing

magnetic flux going through a circuit (so, the first derivative of a

magnetic flux is

__positive__), generates a

*secondary EMF*in

the circuit that is proportional to a rate of change of a magnetic flux,

which, in turn, is proportional to a rate of change of the electric

current in a loop.

So,

**generated (induced) secondary**.

*EMF*is proportional to a rate of change of an electric current in a loopThis induced

*secondary EMF*causes the secondary electric current in a loop that interferes with the original electric current.

The process of interference of two electric currents, varying original

one and that caused by a change of magnetic field flux, is called

**self-induction**.

Now let's talk about the direction of that secondary current from the position of the Law of Energy Conservation.

If the secondary current flows in the same direction as the increasing

original one, the rate of change of electric current in a loop will

increase, causing a greater rate of change of the magnetic flux, causing

even greater rate of change of the electric current flowing through a

wire, causing even greater flux etc. This is a perfect source of free

energy, so it cannot be true.

Indeed, the secondary electric current must be directed opposite to an

increasing original one, thus slowing the rate of increase.

**The**.

*EMF*generated by*self-induction*is opposite to the one that originated the increase in the electric current in a loopAs a practical implication of this principle, when we switch a lamp on,

and the time a switch actually closes the circuit is, for example 0.01

sec, the electric current in a circuit will not reach its maximum during

this exact time, but later, like during 0.02 sec, because of

self-induction that slows the increase of amperage.

Let's consider the process of decreasing of the electric current in a

loop, like when we disconnect a wire from a primary source of

electricity.

Decreasing electric current in a wire loop causes proportional decrease

of a magnetic field intensity at any point around a wire.

This, in turn, causes proportional decrease of

*magnetic flux*flowing through a wire loop with rate of change (first derivative of flux by time)

__negative__.

Negative rate of change of magnetic flux causes the generation of

*secondary (induced) EMF*in a wire, that results in a secondary electric current, that interferes with the decreasing primary one.

Let's analyze the direction of this secondary electric current.

Above, when the primary electric current was increasing and the rate of change of magnetic flux was

__positive__,

we concluded that induced EMF causes the secondary electric current to

go in the opposite to primary direction to preserve the Law of Energy

Conservation.

In a case of decreasing primary electric current we see that the

magnetic flux flowing through a wire loop is decreasing, it's rate of

change (first derivative) is

__negative__, so the

*secondary (induced) EMF*must be directed opposite to the one generated by an increasing primary electric current.

Therefore, if in the case of increasing primary electric current the

*secondary EMF*"worked"

__against the electric current__,

generating a secondary current in a direction opposite to an increasing

primary one, in the case of decreasing primary current the

*secondary (induced) EMF*generates the secondary current

__in the same direction as a decreasing primary current__, "helping" the decreasing primary current, prolonging the decrease.

So, in both cases the

*secondary (induced) EMF*works

__opposite to a rate of change of the magnetic flux__going through a wire loop.

When the magnetic flux increases, the

*induced EMF*generates an

electric current in the opposite direction to a primary one, thus

slowing the rate of increase of the electric current and, therefore,

against the rate of increase of the magnetic flux.

When the magnetic flux decreases, the

*induced EMF*generates an

electric current in the same direction as a primary one, thus slowing

the decrease of the electric current and, therefore, against the rate of

decrease of the magnetic flux.

The above considerations are the primary reason why the Faraday's Law is expressed as

**U = −**d**Φ/**d**t**where the minus sign signifies that the

*induced EMF*works against the rate of change of a magnetic flux.

## No comments:

Post a Comment