Ministry of Education and Science of the Russian Federation
Federal State Budgetary Educational Institution
Higher professional education
National Mineral Resources University “Mining”
Department of General and Technical Physics
(electromagnetism laboratory)
Magnetic Field Study
(Biot-Savart-Laplace law)
Guidelines for laboratory work No. 4
For students of all specialties
SAINT PETERSBURG
Goal of the work: Measurement of magnetic fields created by conductors of various configurations. Experimental verification of the Biot-Savart-Laplace law.
Theoretical foundations of laboratory work
The use of magnetic fields in industry has found wide application. The problem of transmitting energy to certain industrial and other installations can be solved using a magnetic field (for example, in transformers). In the enrichment industry, separation is carried out using a magnetic field (magnetic separators), i.e. separate minerals from waste rock. And during the production of artificial abrasives, ferrosilicon present in the mixture settles to the bottom of the furnace, but small amounts of it are embedded in the abrasive and are later removed by a magnet. Without a magnetic field, electric machine generators and electric motors would not be able to operate. Thermonuclear fusion, magnetodynamic generation of electricity, acceleration of charged particles in synchrotrons, lifting of sunken ships, etc. - all these are areas where magnets are required. Natural magnets, as a rule, are not effective enough in solving some production problems and are mainly used only in household appliances and measuring equipment. The main application of the magnetic field is in electrical engineering, radio engineering, instrument making, automation and telemechanics. Here, ferromagnetic materials are used for the manufacture of magnetic circuits, relays and other magnetoelectric devices. Natural (or natural) magnets occur in nature in the form of deposits of magnetic ores. In mining, separate sections are devoted to the development of magnetic ore deposits and have their own specifics, for example, there are sciences such as magnetochemistry and magnetic flaw detection. The largest known natural magnet is located at the University of Tartu. Its mass is 13 kg and it is capable of lifting a load of 40 kg. The problem of creating strong magnetic fields has become one of the main ones in modern physics and technology. Strong magnets can be created by conductors carrying current. In 1820, G. Oersted (1777–1851) discovered that a current-carrying conductor acts on a magnetic needle, turning it. Just a week later, Ampere showed that two parallel conductors with current in the same direction are attracted to each other. Later, he suggested that all magnetic phenomena are caused by currents, and the magnetic properties of permanent magnets are associated with currents constantly circulating inside these magnets. This assumption is fully consistent with modern ideas. The magnetic field of direct currents of various shapes was studied by the French scientists J. Biot (1774 - 1862) and F. Savard (1791 - 1841). The results of these experiments were summarized by the outstanding French mathematician and physicist P. Laplace. The Bio-Savart-Laplace law, together with the superposition principle, allows us to calculate the magnetic fields created by any current-carrying conductors.
Studying the patterns of magnetic phenomena will allow us to generalize the acquired knowledge and successfully use it both in laboratory conditions and in production.
Magnetic field of a straight conductor carrying current
A conductor through which electric current flows creates a magnetic field. The magnetic field is characterized by the intensity vector `H(Fig. 1), which can be calculated using the formula
`H= òd `H.
According to the Biot-Savart-Laplace law,
Where I– current strength in the conductor, d`l– a vector having the length of an elementary segment of a conductor and directed in the direction of the current, `r– radius vector connecting the element with the point in question P.
Let us consider the magnetic field created by a straight conductor carrying a current of finite length (Fig. 2). Individual elementary sections of this conductor create fields d `H, directed in one direction (perpendicular to the plane of the drawing), therefore the magnetic field strength at point P can be found by integration:
We have l= r o ×сtga, so Moreover, Therefore
Let us calculate the field created by a current flowing through a thin straight wire of infinite length.
Magnetic field induction at an arbitrary point A(Fig. 6.12) created by the conductor element d l , will be equal
Rice. 6.12. Magnetic field of a straight conductor
Fields from different elements have the same direction (tangential to a circle with radius R, lying in a plane orthogonal to the conductor). This means we can add (integrate) absolute values
|
|
Let's express r and sin through the integration variable l
|
|
Then (6.7) can be rewritten as
Thus,
The picture of the magnetic field lines of an infinitely long straight conductor carrying current is shown in Fig. 6.13.
Rice. 6.13. Magnetic field lines of a straight conductor carrying current:
1 - side view; 2, 3 - section of the conductor by a plane perpendicular to the conductor
Rice. 6.14. Designations for the direction of current in a conductor
To indicate the direction of current in a conductor perpendicular to the plane of the figure, we will use the following notation (Fig. 6.14):
Let us recall the expression for the electric field strength of a thin thread charged with a linear charge density
The similarity of expressions is obvious: we have the same dependence on the distance to the thread (current), the linear charge density has been replaced by current strength. But the directions of the fields are different. For a thread, the electric field is directed along the radii. The magnetic field lines of an infinite rectilinear conductor carrying current form a system of concentric circles surrounding the conductor. The directions of the power lines form a right-handed system with the direction of the current.
In Fig. Figure 6.15 presents an experiment in studying the distribution of magnetic field lines around a straight conductor carrying current. A thick copper conductor is passed through holes in a transparent plate on which iron filings are poured. After turning on a direct current of 25 A and tapping on the plate, the sawdust forms chains that repeat the shape of the magnetic field lines.
Around a straight wire perpendicular to the plate, ring lines of force are observed, located most densely near the wire. As you move away from it, the field decreases.
Rice. 6.15. Visualization of magnetic field lines around a straight conductor
In Fig. Figure 6.16 presents experiments to study the distribution of magnetic field lines around wires crossing a cardboard plate. Iron filings poured onto the plate are aligned along the magnetic field lines.
Rice. 6.16. Distribution of magnetic field lines
near the intersection of one, two or several wires with a plate
Let's consider a straight conductor (Fig. 3.2), which is part of a closed electrical circuit. According to the Biot-Savart-Laplace law, the magnetic induction vector 
field created at a point A element 
current carrying conductor I,
has the meaning 
, Where 
- angle between vectors 
And 
. For all areas 
this conductor vectors 
And 
lie in the plane of the drawing, therefore at the point A all vectors 
, created by each section 
, directed perpendicular to the plane of the drawing (towards us). Vector 
determined by the principle of field superposition:
,
its module is equal to:
.
Let us denote the distance from the point A to the conductor 
. Consider a conductor section 
. From point A let's draw an arc WITHD radius 
,
– small, therefore 
And 
. From the drawing it is clear that 
;
, But 
(CD=
) Therefore we have:
.
For 
we get:
Where 
And 
- angle values for the extreme points of the conductor MN.
If the conductor is infinitely long, then 
,
. Then
the induction at each point of the magnetic field of an infinitely long straight conductor with current is inversely proportional to the shortest distance from this point to the conductor.
3.4. Magnetic field of circular current
Consider a circular turn of radius R, through which current flows I
(Fig. 3.3) .
According to the Biot-Savart-Laplace law, induction 
field created at a point ABOUT element 
turn with current is equal to:
,
and 
, That's why 
, And 
. Taking this into account, we get:
.
All vectors 
directed perpendicular to the drawing plane towards us, therefore induction
tension 
.
Let S– area covered by a circular turn, 
. Then the magnetic induction at an arbitrary point on the axis of a circular coil with current:
,
Where 
– distance from the point to the surface of the coil. It is known that 
- magnetic moment of the turn. Its direction coincides with the vector 
at any point on the axis of the coil, therefore 
, And 
.
Expression for 
similar in appearance to the expression for the electric displacement at field points lying on the axis of the electric dipole sufficiently far from it:
.
Therefore, the magnetic field of the ring current is often considered as the magnetic field of some conventional “magnetic dipole”; the positive (north) pole is considered to be the side of the plane of the coil from which the magnetic field lines exit, and the negative (south) pole is the one into which they enter.
For a current loop of arbitrary shape:
,
Where 
- unit vector of outer normal to the element 
surfaces S,
limited by a contour. In the case of a flat contour, the surface S
– flat and all vectors 
match up.
3.5. Solenoid magnetic field
A solenoid is a cylindrical coil with a large number of turns of wire. The solenoid turns form a helical line. If the turns are located closely, then the solenoid can be considered as a system of series-connected circular currents. These turns (currents) have the same radius and a common axis (Fig. 3.4).
Let's consider the cross section of the solenoid along its axis. We will use circles with a dot to denote currents coming from behind the drawing plane towards us, and a circle with a cross will denote currents coming beyond the drawing plane, away from us. L– solenoid length, n
–
number of turns per unit length of the solenoid; -
R- radius of the turn. Consider the point A, lying on the axis 
solenoid. It is clear that magnetic induction 
at this point is directed along the axis 
and is equal to the algebraic sum of the inductions of magnetic fields created at this point by all turns.
Let's draw from the point A radius – vector 
to any turn. This radius vector forms with the axis 
corner α
. The current flowing through this turn creates at the point A magnetic field with induction 
.
Let's consider a small area 
solenoid, it has 
turns. These turns are created at a point A magnetic field, the induction of which
.
It is clear that the axial distance from the point A to the site 
equals 
; Then 
.Obviously, 
, Then
Magnetic induction of fields created by all turns at a point A equal to
Magnetic field strength at a point A
.
From Fig. 3. 4 we find: 
;
.
Thus, magnetic induction depends on the position of the point A on the solenoid axis. She
maximum in the middle of the solenoid:
.
If L>>
R, then the solenoid can be considered infinitely long, in this case 
,
,
,
; Then
;
.
At one end of the long solenoid 
,
or 
;
,
,
.
An electric current in a conductor produces a magnetic field around the conductor. Electric current and magnetic field are two inseparable parts of a single physical process. The magnetic field of permanent magnets is ultimately also generated by molecular electric currents formed by the movement of electrons in orbits and their rotation around their axes.
The magnetic field of a conductor and the direction of its lines of force can be determined using a magnetic needle. The magnetic lines of a straight conductor have the shape of concentric circles located in a plane perpendicular to the conductor. The direction of magnetic field lines depends on the direction of the current in the conductor. If the current in the conductor comes from the observer, then the lines of force are directed clockwise.
The dependence of the direction of the field on the direction of the current is determined by the gimlet rule: when the translational movement of the gimlet coincides with the direction of the current in the conductor, the direction of rotation of the handle coincides with the direction of the magnetic lines.
The gimlet rule can also be used to determine the direction of the magnetic field in the coil, but in the following formulation: if the direction of rotation of the gimlet handle is combined with the direction of the current in the turns of the coil, then the translational movement of the gimlet will show the direction of the field lines inside the coil (Fig. 4.4).
Inside the coil these lines go from the south pole to the north, and outside it - from north to south.
The gimlet rule can also be used to determine the direction of current if the direction of the magnetic field lines is known.
A current-carrying conductor in a magnetic field experiences a force equal to
F = I·L·B·sin
I is the current strength in the conductor; B - module of the magnetic field induction vector; L is the length of the conductor located in the magnetic field; is the angle between the magnetic field vector and the direction of the current in the conductor.
The force acting on a current-carrying conductor in a magnetic field is called the Ampere force.
The maximum ampere force is:
F = I L B
The direction of the Ampere force is determined by the left hand rule: if the left hand is positioned so that the perpendicular component of the magnetic induction vector B enters the palm, and four extended fingers are directed in the direction of the current, then the thumb bent 90 degrees will show the direction of the force acting on the segment conductor with current, that is, Ampere force.
If and lie in the same plane, then the angle between and is straight, therefore . Then the force acting on the current element is
(of course, from the side of the first conductor, exactly the same force acts on the second).
The resulting force is equal to one of these forces. If these two conductors influence the third, then their magnetic fields need to be added vectorially.
Circuit with current in a magnetic field
| Rice. 4.13 |
Let a frame with current be placed in a uniform magnetic field (Fig. 4.13). Then the Ampere forces acting on the sides of the frame will create a torque, the magnitude of which is proportional to the magnetic induction, the current strength in the frame, and its area S and depends on the angle a between the vector and the normal to the area:
The normal direction is chosen so that the right screw moves in the normal direction when rotating in the direction of the current in the frame.
The maximum value of the torque is when the frame is installed perpendicular to the magnetic lines of force:
This expression can also be used to determine the magnetic field induction:
A value equal to the product is called the magnetic moment of the circuit R t. The magnetic moment is a vector whose direction coincides with the direction of the normal to the contour. Then the torque can be written
At angle a = 0 the torque is zero. The value of the torque depends on the area of the contour, but does not depend on its shape. Therefore, any closed circuit through which direct current flows is subject to a torque M, which rotates it so that the magnetic moment vector is parallel to the magnetic field induction vector.
Good day to all. In the last article I talked about the magnetic field and dwelled a little on its parameters. This article continues the topic of the magnetic field and is devoted to such a parameter as magnetic induction. To simplify the topic, I will talk about the magnetic field in a vacuum, since different substances have different magnetic properties, and as a result, it is necessary to take their properties into account.
Biot–Savart–Laplace law
As a result of studying the magnetic fields created by electric current, researchers came to the following conclusions:
- magnetic induction created by electric current is proportional to the strength of the current;
- magnetic induction depends on the shape and size of the conductor through which the electric current flows;
- magnetic induction at any point in the magnetic field depends on the location of this point in relation to the current-carrying conductor.
The French scientists Biot and Savard, who came to such conclusions, turned to the great mathematician P. Laplace to generalize and derive the basic law of magnetic induction. He hypothesized that the induction at any point of the magnetic field created by a current-carrying conductor can be represented as the sum of the magnetic inductions of elementary magnetic fields that are created by an elementary section of a current-carrying conductor. This hypothesis became the law of magnetic induction, called Biot-Savart-Laplace law. To consider this law, let us depict a current-carrying conductor and the magnetic induction it creates
Magnetic induction dB created by an elementary section of a conductor dl.
Then magnetic induction dB elementary magnetic field that is created by a section of a conductor dl, with current I at an arbitrary point R will be determined by the following expression
where I is the current flowing through the conductor,
r is the radius vector drawn from the conductor element to the magnetic field point,
dl is the minimum conductor element that creates induction dB,
k – proportionality coefficient, depending on the reference system, in SI k = μ 0 /(4π)
Because is a vector product, then the final expression for the elementary magnetic induction will look like this
Thus, this expression allows us to find the magnetic induction of the magnetic field, which is created by a conductor with a current of arbitrary shape and size by integrating the right side of the expression
where the symbol l indicates that integration occurs along the entire length of the conductor.
Magnetic induction of a straight conductor
As you know, the simplest magnetic field creates a straight conductor through which electric current flows. As I already said in the previous article, the lines of force of a given magnetic field are concentric circles located around the conductor.
To determine magnetic induction IN straight wire at a point R Let us introduce some notation. Since the point R is at a distance b from the wire, then the distance from any point on the wire to the point R is defined as r = b/sinα. Then the shortest length of the conductor dl can be calculated from the following expression
As a result, the Biot–Savart–Laplace law for a straight wire of infinite length will have the form
where I is the current flowing through the wire,
b is the distance from the center of the wire to the point at which the magnetic induction is calculated.
Now we simply integrate the resulting expression over dα ranging from 0 to π.
Thus, the final expression for the magnetic induction of a straight wire of infinite length will have the form
I – current flowing through the wire,
b is the distance from the center of the conductor to the point at which the induction is measured.
Magnetic induction of the ring
The induction of a straight wire has a small value and decreases with distance from the conductor, therefore it is practically not used in practical devices. The most widely used magnetic fields are those created by a wire wound around a frame. Therefore, such fields are called magnetic fields of circular current. The simplest such magnetic field is possessed by an electric current flowing through a conductor, which has the shape of a circle of radius R.
In this case, two cases are of practical interest: the magnetic field at the center of the circle and the magnetic field at point P, which lies on the axis of the circle. Let's consider the first case.
In this case, each current element dl creates an elementary magnetic induction dB in the center of the circle, which is perpendicular to the contour plane, then the Biot-Savart-Laplace law will have the form
All that remains is to integrate the resulting expression over the entire length of the circle
where μ 0 is the magnetic constant, μ 0 = 4π 10 -7 H/m,
I – current strength in the conductor,
R is the radius of the circle into which the conductor is rolled.
Let's consider the second case, when the point at which the magnetic induction is calculated lies on the straight line X, which is perpendicular to the plane limited by the circular current.
In this case, induction at the point R will be the sum of elementary inductions dB X, which in turn is a projection onto the axis X elementary induction dB
Applying the Biot-Savart-Laplace law, we calculate the value of magnetic induction
Now let’s integrate this expression over the entire length of the circle
where μ 0 is the magnetic constant, μ 0 = 4π 10 -7 H/m,
I – current strength in the conductor,
R is the radius of the circle into which the conductor is rolled,
x is the distance from the point at which the magnetic induction is calculated to the center of the circle.
As can be seen from the formula for x = 0, the resulting expression transforms into the formula for magnetic induction at the center of the circular current.
Circulation of the magnetic induction vector
To calculate the magnetic induction of simple magnetic fields, the Biot-Savart-Laplace law is sufficient. However, with more complex magnetic fields, for example, the magnetic field of a solenoid or toroid, the number of calculations and the cumbersomeness of the formulas will increase significantly. To simplify calculations, the concept of circulation of the magnetic induction vector is introduced.
Let's imagine some contour l, which is perpendicular to the current I. At any point R of this circuit, magnetic induction IN directed tangentially to this contour. Then the product of vectors dl And IN is described by the following expression
Since the angle dφ small enough, then the vectors dl B defined as arc length
Thus, knowing the magnetic induction of a straight conductor at a given point, we can derive an expression for the circulation of the magnetic induction vector
Now it remains to integrate the resulting expression over the entire length of the contour
In our case, the magnetic induction vector circulates around one current, but in the case of several currents, the expression for the circulation of magnetic induction turns into the law of total current, which states:
The circulation of the magnetic induction vector in a closed loop is proportional to the algebraic sum of the currents that the given loop covers.
Magnetic field of solenoid and toroid
Using the law of total current and circulation of the magnetic induction vector, it is quite easy to determine the magnetic induction of such complex magnetic fields as those of a solenoid and a toroid.
A solenoid is a cylindrical coil that consists of many turns of conductor wound turn to turn on a cylindrical frame. The magnetic field of a solenoid actually consists of multiple magnetic fields of a circular current with a common axis perpendicular to the plane of each circular current.
Let's use the circulation of the magnetic induction vector and imagine the circulation along a rectangular contour 1-2-3-4 . Then the circulation of the magnetic induction vector for a given circuit will have the form
Since in the areas 2-3 And 4-1 the magnetic induction vector is perpendicular to the circuit, then the circulation is zero. Location on 3-4 , which is significantly removed from the solenoid, then it can also be ignored. Then, taking into account the law of total current, the magnetic induction in a solenoid of sufficiently large length will have the form
where n is the number of turns of the solenoid conductor per unit length,
I – current flowing through the solenoid.
A toroid is formed by winding a conductor around a ring frame. This design is equivalent to a system of many identical circular currents, the centers of which are located on a circle.
As an example, consider a toroid of radius R, on which it is wound N turns of wire. Around each turn of the wire we take a radius contour r, the center of this contour coincides with the center of the toroid. Since the magnetic induction vector B is directed tangentially to the contour at each point of the contour, then the circulation of the magnetic induction vector will have the form
where r is the radius of the magnetic induction loop.
The circuit passing inside the toroid covers N turns of wire with current I, then the law of the total current for the toroid will have the form
where n is the number of turns of the conductor per unit length,
r – radius of the magnetic induction loop,
R is the radius of the toroid.
Thus, using the law of total current and the circulation of the magnetic induction vector, it is possible to calculate an arbitrarily complex magnetic field. However, the law of total current gives correct results only in a vacuum. When calculating magnetic induction in a substance, it is necessary to take into account the so-called molecular currents. This will be discussed in the next article.
Theory is good, but without practical application it is just words.