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

__Magnetic Flux__

Consider a flat lying completely within an XY-coordinate plane closed

wire loop that includes a battery with direct electric current

*running through it.*

**I**Since electric current generates a magnetic field around each however small segment of a wire

*d*

**S**with circular magnetic field lines around it in the plane perpendicular

to this segment of a wire, the general direction of all magnetic lines

inside a wire loop will be parallel to Z-axis.

More precisely, vectors of magnetic field intensity

*at*

**B**all points of XY-plane inside a wire loop are similarly directed

perpendicularly to this plane and parallel to Z-axis. They are not

necessarily equal in magnitude, though.

Consider an infinitesimal two-dimensional area of XY-plane inside a wire loop

*d*around some point

**A***.*

**P**Since it's infinitesimal, we can assume that the intensity of the

magnetic field inside it is uniform and equal to the intensity at point

*, which we will denote as vector*

**P***.*

**B(P)**This intensity is a result of combined magnetic field generated at that point by all infinitesimal segments

*d*of a wire.

**S**Each one of these components of a resulting intensity vector has a

direction perpendicular to a direction of electric current in segment

*d*(its source) and to a vector

**S***from a chosen infinitesimal segment*

**R***d*of a wire loop to point

**S***, from which follows the perpendicularity of all vectors of magnetic field intensity to XY-plane and parallel to Z-axis.*

**P**Magnetic field, as other force fields, is

*additive*, that is, the

combined intensity vector generated by an entire wire loop is a vector

sum of intensity vectors generated by each segment of a wire.

All these individual magnetic fields generated by different segments of a

wire are directed along the Z-axis, so we will concentrate only on

their magnitude to find the result of their addition.

Knowing the geometry of a wire and the electric current

*in it, we can use the law we described in the previous lecture that determines the magnetic field intensity at any point*

**I***inside a wire loop generated by its any infinitesimal segment*

**P***d*:

**S***d*

**B = μ**d_{0}·I·**S·sin(α) / (4πR²)**where

*is a magnitude of a vector*

**R***from a chosen infinitesimal segment*

**R***d*of a wire loop to point

**S***,*

**P***d*is the infinitesimal segment of a wire,

**S***is an angle between the direction of electric current in the segment*

**α***d*and a vector

**S***from this segment to a point*

**R***,*

**P***is the*

**μ**_{0}*permeability*of free space.

Integrating this along the entire wire (this is integration along a curve) gives the value of magnetic field intensity

*at point*

**B(P)***.*

**P**Now we assume that this process is done and function

*is determined, that is we know the magnitude*

**B(P)***of the intensity vector of magnetic field generated by our wire loop at any point*

**B(P)***inside a loop.*

**P**Recall that the direction of this vector is always along Z-axis, that is perpendicular to a plane of a wire loop.

We define

*magnetic field flux*

*as a two-dimensional integral of a product of intensity*

**Φ***by area*

**B(P)***d*.

**A**In a simple case of a constant magnetic field intensity

*at any point*

**B(P)=B***inside a loop this is a simple product of a constant magnetic field intensity*

**P***by the area of a wire loop*

**B**

**A**

**Φ = B · A**If

*is variable inside a loop, the two-dimensional integration by area of the loop produces the*

**B(P)***magnetic field flux*

*∫∫*

**Φ =**_{A}

**B(P)·**d**A**The complexity of exact calculation of magnetic field intensity

*at any point*

**B(P)***inside a loop and subsequent calculation of the*

**P***magnetic field flux*

flowing through the wire loop are beyond the scope of this course, so

in our practice problems we will usually assume the uniformity of the

magnetic field with constant

*.*

**B(P)=B**A concept of

*magnetic field flux*is applicable not only for a

magnetic field generated by a wire loop with electric current running

through it. It is a general concept used to characterize the combined

effect of a magnetic field onto the area where we observe it.

Consider an analogy:

*magnetic field intensity*and amount of grass growing in the backyard in a season per unit of area,

*wire loop area*and area of a backyard,

*magnetic field flux*and how much grass growth in an entire backyard in a season.

Another analogy:

*magnetic field intensity*and productivity of each individual doing some work,

*wire loop area*and a team of people doing this work,

*magnetic field flux*and productivity of a team doing this work.

A general

*magnetic field flux*is defined for a given magnetic field with known intensity vector

*at each point*

**B(P)***of space and given two-dimensional finite surface*

**P***in space, through which magnetic field lines are going.*

**S**The purpose of our definition of

*magnetic field flux*in this general case is to quantify the total amount of "magnetic field energy" flowing through a surface

*.*

**S**Notice a small area around point

*on a picture above.*

**P**If we consider it to be infinitesimally small, we can assume that

magnetic field intensity at any point of this small area is the same as

at point

*.*

**P**If this small area is perpendicular to vector

*, we would just multiply the magnitude of intensity vector*

**B(P)***in*

**B(P)***tesla*by the area Δ

*of a small area around point*

**A***in*

**P***square meters*and obtain the magnetic field flux Δ

*going through this small area in units of magnetic flux -*

**Φ***weber = tesla · meter²*.

In a more general case, when this small area around point

*is not perpendicular to magnetic field intensity vector*

**P***, to obtain the*

**B(P)***magnetic field flux*going through it, we have to multiply it by

*, where*

**cos(φ)***is an angle between magnetic field intensity vector*

**φ(P)***and normal (perpendicular vector) to a surface at point*

**B(P)***.*

**P**So, the expression for an infinitesimal

*magnetic field flux*around point

*is*

**P***d*

**Φ = B(P)·cos(φ(P))·**d**A**The unit of measurement of

*magnetic field flux*is

**Weber (Wb)**that is equal to a flux of a magnetic field of intensity through an area of

**1 tesla (1T)**

**1 square meter (1m²)**:

**1 Wb = 1 T · 1 m²**Once we have determined the

*magnetic field flux*flowing through a

small area around each point of a surface, we just have to summarize

all these individual amounts to obtain the total

*magnetic field flux*flowing through an entire surface.

Mathematically, it means we have to integrate the above expression for

*d*along a surface

**Φ***.*

**S**We will not consider this most general case in this course. Our magnetic fields will be, mostly, uniform and surfaces flat.

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