MAGNETIC INDUCTION: FORMULAS, HOW IT IS CALCULATED AND EXAMPLES - PHYSICAL

The magnetic induction or magnetic flux density is altered the environment caused by the presence of electrical currents. They modify the nature of the space that surrounds them, creating a vector field.

The vector magnetic induction, magnetic flux density or simply magnetic field B, has three distinctive characteristics: an intensity expressed by a numerical value, a direction and also a sense given at each point in space. It is highlighted in bold to distinguish it from purely numerical or scalar quantities.

Rule of the right thumb to determine the direction and sense of the magnetic induction vector. Source: Jfmelero

The right thumb rule is used to find the direction and direction of the magnetic field caused by a current carrying wire, as shown in the figure above.

The thumb of the right hand should point in the direction of the current. Then the rotation of the four remaining fingers indicates the shape of B, which in the figure is represented by the concentric red circles.

In such a case, the direction of B is tangential to the circumference concentric with the wire and the direction is counterclockwise.

The magnetic induction B in the International System is measured Tesla (T), however it is more frequent to measure it in another unit called Gauss (G). Both units were named respectively in honor of Nikola Tesla (1856-1943) and Carl Friedrich Gauss (1777-1855) for their extraordinary contributions to the science of electricity and magnetism.

What are the properties of magnetic induction or magnetic flux density?

A compass placed near live wire will always line up with B. Danish physicist Hans Christian Oersted (1777-1851) was the first to notice this phenomenon in the early 19th century.

And when the current stops, the compass points to geographic north again, as always. By carefully changing the position of the compass, you get a map of the shape of the magnetic field.

This map is always in the shape of circles concentric to the wire, as described at the beginning. In this way, B.

Even if the wire is not straight, vector B will form concentric circles around it. To determine the shape of the field, just imagine very small segments of wire, so small that they appear rectilinear and surrounded by concentric circles.

Magnetic field lines produced by a current-carrying loop of wire. Source: Pixabay.com

This points to an important property of magnetic field lines B: they have no beginning or end, they are always closed curves.

Biot-Savart's law

The 19th century marked the beginning of the age of Electricity and Magnetism in science. 1820 near the French physicists Jean Marie Biot (1774-1862) and Felix Savart (1791-1841) discovered the law that bears his name and that calculates the vector B.

They made the following observations about the contribution to the magnetic field produced by a wire segment of differential length dl carrying an electric current I:

The magnitude of B decreases with the inverse of the square of the distance to the wire (this makes sense: away from the wire the intensity of B must be less than at nearby points).
The magnitude of B is proportional to the intensity of the current I that passes through the wire.
The direction of B is tangential to the circle of radius r centered on the wire and the direction of B is given, as we said, by the rule of the right thumb.

The cross product or cross product is the appropriate mathematical tool to express the last point. To establish a vector product, two vectors are needed, which are defined as follows:

d l is the vector whose magnitude is the length of the differential segment dl
r is the vector that goes from the wire to the point where you want to find the field

Formulas

All of this can be combined into a mathematical expression:

The constant of proportionality necessary to establish equality is the magnetic permeability of free space μ _o = 4π.10 ^-7 Tm / A

This expression is the Biot and Savart law, which allows us to calculate the magnetic field of a current segment.

Such a segment in turn must be part of a larger and more closed circuit: a current distribution.

The condition that the circuit is closed is necessary for an electric current to flow. Electric current cannot flow in open circuits.

Finally, to find the total magnetic field of said current distribution, all the contributions of each differential segment d l are added. This is equivalent to integrating over the entire distribution:

To apply the Biot-Savart law and calculate the magnetic induction vector, it is necessary to consider some very important important points:

The cross product between two vectors always results in another vector.
It is convenient to find the vector product before proceeding to the resolution of the integral, then the integral of each of the components obtained separately is solved.
It is necessary to draw a picture of the situation and establish a suitable coordinate system.
Whenever the existence of some symmetry is observed, it should be used to save calculation time.
When there are triangles, the Pythagorean theorem and the cosine theorem are helpful in establishing the geometric relationship between the variables.

How is it calculated?

With a practical example of the calculation of B for a straight wire, these recommendations apply.

Example

Calculate the magnetic field vector that a very long rectilinear wire produces at a point P in space, according to the figure shown.

Geometry necessary to calculate the magnetic field at point P, of an infinitely long current wire. Source: self made.

From the figure you have to:

The wire is directed in a vertical direction, with current I flowing upward. This direction is + y in the coordinate system, whose origin is at point O.
In this case, according to the rule of the right thumb, B at point P is directed towards the inside of the paper, therefore it is denoted by a small circle and an "x" in the figure. This address will be taken as -z.
The right triangle whose legs are y and R, relates both variables according to the Pythagorean theorem: r ² = R ² + y ²

All this is substituted in the integral. The cross product or cross is indicated by its magnitude plus its direction and its sense:

The proposed integral is found in a table of integrals or it is solved by an appropriate trigonometric substitution (the reader can check the result using y = Rtg θ):

The result agrees with what was expected: the magnitude of the field decreases with distance R and increases proportionally with the intensity of current I.

Although an infinitely long wire is an idealization, the expression obtained is a very good approximation for the field of a long wire.

With Biot and Savart's law it is possible to find the magnetic field of other highly symmetric distributions, such as a circular loop that carries current, or bent wires combining rectilinear and curvilinear segments.

Of course, to analytically solve the proposed integral, the problem must have a high degree of symmetry. Otherwise the alternative is to solve the integral numerically.

References

Serway, R., Jewett, J. (2008). Physics for Science and Engineering. Volume 2. Mexico. Cengage Learning Editors. 367-372.

MAGNETIC INDUCTION: FORMULAS, HOW IT IS CALCULATED AND EXAMPLES - PHYSICAL - 2025