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

__Magnetism of Electric Current in a Loop__

Let's recall some general considerations of any field produced by a

point-source (like gravitational or electrostatic) and the force it

exerts at certain distance from its center.

In three-dimensional space the field influence is spreading around in

all directions from its center, and at any given time it reaches and

spreads over a sphere of the area

*, where*

**4πR²***is the radius of this sphere.*

**R**This important consideration prompted the dependency between the

magnitude of the field force at any point in space and inverse of a

square of a distance between this point and the source of a field.

This dependency exists in all laws related to fields we studied, and it's true for magnetic field as well.

There is, however, an important complication with the sources of

magnetic field. Strictly speaking, they are not and theoretically cannot

be just points, because they always have a characteristic of the

direction. If it's a permanent magnet, there is a North pole to South

pole direction. If the source is a direct current, there is a direction

of electrons that causes the existence of a magnetic field.

So, besides the distance from the source of a magnetic field, another

important aspect that influences the magnetic field is how the point of

measuring the magnetic field is positioned relatively to a direction of

the electric current at the source of a field.

Consider measuring a strength of a magnetic field at point

*on the following picture.*

**P**Here an infinitesimal segment

*of the length*

**AB***d*of a wire with an electric current

**S***produces a magnetic field that we are measuring at point*

**I***. From the viewpoint of point*

**P***segment*

**P***is visible at an angle and seems having the size of a segment*

**AB***that is perpendicular to a radius-vector*

**BC***from the source of magnetic field segment*

**R***to the point of measuring*

**AB***.*

**P**Simple geometry gives the visible from point

*length of segment*

**P***- the length of*

**AB***, which is the length of*

**BC***multiplied by*

**AB***.*

**sin(α)**With this correction to a simple dependency of the strength of magnetic

field on the inverse square of a distance from the field source we can

represent the magnitude of the vector of magnetic field strength or

intensity at point

*as*

**P***d*

**B = k·I·**d**S·sin(α) / R²**The constant

*in SI units equals to*

**k***, where*

**μ**_{0}/(4π)*is the*

**μ**_{0}*permeability*of free space. So the formula for a magnitude of the strength of magnetic field looks like this:

*d*

**B = μ**d_{0}·I·**S·sin(α) / (4πR²)**The vector of force has magnitude and direction. The magnitude was

evaluated above. The direction of this force, based on experimental

data, is always perpendicular to both the direction of the current

*and the direction of the radius-vector from the source of magnetic field to a point of measurement*

**I***.*

**P**This allows us to express the vector of magnetic field intensity

*in a form of a vector product*

**B***⨯*

**B = μ**_{0}·I·dS

**r / (4πR²)**where

*is a unit vector in the direction from the source of magnetic field segment*

**r***to the point of measurement*

**AB***.*

**P**Equivalent formula is

*⨯*

**B = μ**_{0}·I·dS

**R / (4πR³)**Let's apply this formula to calculate the magnetic field intensity and direction in a center of a circular wire loop of radius

*with electric current*

**R***running through it.*

**I**We divide the circular wire loop into infinitesimal segments of the length

*d*. Fortunately, all radiuses from any such segment to a center of a loop are perpendicular to these segments, so

**S***in the above formula for a magnitude of the intensity vector.*

**sin(α)=1**The magnetic field intensity

*from each segment equals to*

**B***d*

**B = μ**d_{0}·I·**S / (4πR²)**The direction of this force is perpendicular to both radius to a segment

*d*

**S**and to a segment itself. That is, it should be perpendicular to a plane

of the wire loop, that is the vector of magnetic field force in a

center of a wire loop is always directed perpendicularly to the loop.

Since it's true for any segment

*d*, we can just add all the forces from all the segments, that is, we have to integrate the above expression by

**S***along the whole circle from*

**S***to*

**0***.*

**2πR**The only variable in the above expression is

*d*, so the result is

**S***∫*

**B =**_{[0,2πR]}

**μ**d_{0}·I·**S/(4πR²) =**

= 2πR·μ= 2πR·μ

_{0}·I/(4πR²)= μ_{0}·I/(2R)
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