Modern Physics

Magnetic induction: formulas, how it is calculated and examples

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

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

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

In this case, the direction of B is tangential to the concentric circle with the wire and the direction is counterclockwise.

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

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

A compass is placed near the wire chain and is always aligned 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 back to geographic north, as usual. By carefully changing the position of the compass, a map of the shape of the magnetic field is obtained.

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This map always takes the form of concentric circles in relation to the wire, as described at the beginning. That way you can display 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 straight and surrounded by concentric circles.

This indicates an important property of magnetic field lines B : they have no beginning and no end, they are always sharp curves.

Biot-Savart’s Law

The nineteenth century ushered in the age of electricity and magnetism in science. Circa 1820 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 segment of wire of differential length dl that carries an electric current I :

  • The magnitude of B decreases with the inverse square of the distance from 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 that passes through the wire.
  • The direction of B is tangential to the circumference of the 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 for expressing the last point. Two vectors are needed to establish a cross product, defined as follows:

  • 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.
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formulas

All of this can be combined into a mathematical expression

Magnetic induction: formulas, how it is calculated and examples 3
The proportionality constant necessary to establish equality is the magnetic permeability of the free space μ o = 4π.10 -7 Tm / A

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

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

The closing condition of the circuit is necessary for electrical current to flow. Electric current cannot flow in open circuits.

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

Magnetic induction: formulas, how it is calculated and examples 3

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.

Magnetic induction: formulas, how it is calculated and examples 3

  • It is convenient to find the cross product before moving on to solve 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 an appropriate coordinate system.
  • Whenever the existence of any symmetry is noted, it should be used to save calculation time.
  • When there are triangles, the Pythagorean theorem and the cosine are of great help in establishing the geometric relationship between the variables.

How is it calculated?

As a practical example of calculating B for a straight wire, these recommendations apply.

Example

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

Geometry needed to calculate the magnetic field at point P of an infinitely long current wire. Source: own elaboration.

From the figure, you should:

  • The wire is routed in the vertical direction, with current I flowing upwards. This address is + and in the coordinate system, whose origin is at point O.

Magnetic induction: formulas, how it is calculated and examples 3

  • In this case, according to the right thumb rule B at point P is directed towards the paper, so that it is indicated with a small circle and an “x” in the figure. This address will be taken as -z.
  • The triangle whose legs are e and R , both variables related according to Pythagoras’ theorem: 2 = R 2 + y 2

All of this is replaced in full. The vector or cross product is indicated by its magnitude plus its direction and meaning:

Magnetic induction: formulas, how it is calculated and examples 8

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

Magnetic induction: formulas, how it is calculated and examples 9

The result is consistent with expectations: the field magnitude decreases with distance R and increases proportionally with the intensity of current I.

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

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

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

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