Content uploaded by Alexander I. Korolev

Author content

All content in this area was uploaded by Alexander I. Korolev on Oct 20, 2015

Content may be subject to copyright.

On electromagnetic induction in electric conductors

ALEXANDER I. KOROLEV

Department of General Physics I, Faculty of Physics, St. Petersburg State University, 7/9 Universitetskaya nab.,

199034, Saint Petersburg, Russian Federation

e-mail: alex-korolev@ya.ru

phone: +79045517888

Experimental validation of the Faraday’s law of electromagnetic induction (EMI) is performed

when an electromotive force is generated in thin copper turns, located inside a large magnetic

coil. It has been established that the electromotive force (emf) value should be dependent not

only on changes of the magnetic induction flux through a turn and on symmetry of its crossing

by magnetic power lines also. The law of EMI is applicable in sufficient approximation in case of

the changes of the magnetic field near the turn are symmetrical. Experimental study of the

induced emf in arcs and a direct section of the conductor placed into the variable field has been

carried out. Linear dependence of the induced emf on the length of the arc has been

ascertained in case of the magnetic field distribution symmetry about it. Influence of the

magnetic field symmetry on the induced emf in the arc has been observed. The curve of the

induced emf in the direct section over period of current pulse is similar to this one for the turns

and arcs. The general law of EMI for a curvilinear conductor has been deduced. Calculation of

the induced emf in the turns wrapped over it and comparison with the experimental data has

been made. The proportionality factor has been ascertained for the law. Special conditions

have been described, when the induced emf may not exist in the presence of inductive current.

Theoretical estimation of the inductive current has been made at a induced low voltage in the

turn. It has been noted the necessity to take into account the concentration of current carriers

in calculation of the induced emf in semiconductors and ionized conductors.

Introduction

For the first time the phenomenon of induction of electrical current in a conductor under a

variable magnetic field was described by F. Zantedeschied in 1829 and by M. Faraday in 1831

[1]. M. Faraday established that "electrotonic" state in a conductor appears at crossing by

magnetic power lines. Many experiments on quantitative investigation of "magnetoelectric"

induction were carried out. A description that allows detecting the induction current direction

knowing the mode of magnetic power lines movement was given. A brief metaphysical rule of

the direction detecting was formulated by E. Lenz in 1832 [2]. First mathematical expression of

the emi law was presented by F. Neumann in 1845 -1847 [3]. He introduced a concept of

“vector - potential “, expressing it through the induction of magnetic field. Contribution to the

theory of electromagnetism was made by Felici.

Wide theoretical description of electromagnetic phenomena was made by J. Maxwell in the

middle of 19th century [4,5]. The scientist put the experimental results, obtained by M.

Faraday, into the language of mathematics. The expression for components of the emf in a loop

was given as a system of three scalar equations (1).

Here P, Q and R are the components of the emf representing the potential drop per unit of

length of a conductor along x, y, z axes of Cartesian coordinates, respectively, at t point of time.

F, G and H are projections of the electromagnetic momentum on x, y, z axes. ψ is the electrical

potential at the point under consideration. Expression for an induced emf in a fixed loop of

arbitrary shape is

Where dS is an element of the contour of integration. At integration, the terms with ψ are

cancelled. Thus, ξ is defined by the derivatives of the electromagnetic momentum. This is the

result of application of the second Newton's law to the current carriers in the loop.

Further contribution to electromagnetic theory was made by O. Heaviside [6]. He transformed

some Maxwell's equations using the terms of three-dimensional vectors, intensities and

induction of electrical and magnetic fields. The discovery of electromagnetic waves by Hertz

promoted the deduction of two well- known equations of Hertz- Heaviside (now they are called

as rotary Maxwell's equations). There the electric and magnetic fields intensities are related,

irrespective of the presence of particles in the medium that let us to describe the propagation

of electromagnetic waves in some approximation. Nevertheless, it does not mean that one

variable field induces another one [7,8]. In [7] there is a conclusion that the momentum in (1) is

induced by “outside forces” of the variable magnetic field. The expression for these forces via

vector-potential is given in [8].

In classical electrodynamics the law of emi for a closed loop (3) is an integral equivalent for the

first Hertz- Heaviside's equation for vacuum (4) [9, 10].

(4)

According to (3), the induced emf is expressed by a time derivative of the magnetic flux Φ

through a loop (in SI). However, the electric and magnetic fields in a real conductor differ from

these in transverse electromagnetic wave in vacuum and they depend on the conductor

charges and its material. As well as it is shown in [11] the variable magnetic field can propagate

independently of the variable electrical one. Thus, the transfer from (4) to (3) is not correct.

And the paradoxes are known when calculation of the induced emf using (3) without taking

additional reservations [12,13] into account are made. In this connection, the experimental

verification of (3) and formulation of more exact law for emi for a conductor of arbitrary form is

of our interest.

Measurements of magnetic field in the coil using induction sensors

The magnetic coil is a source of the field (see Fig. 1) is winded by a copper bus in textile

isolation with section of 2×4 mm2. Number of turns is 40, the inner diameter of winding is 12

mm, the outer diameter is 35 mm, the length - 40 mm. The winding of the coil enables to make

current outlets far from its axis. A 3 mm textolite lining is inserted into the central part of the

coil to avoid the electrical breakdown (see Fig. 1). The coil is placed inside a cast- iron bandage

and tightened with thick isolating jaws on each side. The coil is intended for generation of a

strong magnetic field but is suitable for generation of weak fields too.

1

2

3

4

5

6

7

Fig. 1. Axial section of the magnetic coil. 1 - coil windings, 2 - induction sensor, 3 - sensor winding, 4 - isolating jaw,

5 - isolating lining, 6 - cast- iron bandage, 7 -bolt .

Induction sensors (2 pcs.) are thin ceramic tubes 2.5 mm and 5.5 mm in diameter with copper

windings at the ends (see Fig. 1). The wire is covered by varnish isolation, the wire diameter is

0.1 mm, the number of turns is 10. The turns are tight to each other forming coils 1 mm in

length. The current outlets of the coils are parallel to the tubes. The sensors are fixed at a

support so that their axes coincide with the axe of the magnetic coil. They are placed into the

coil in this position. The Hall sensor Honeywell SS496A is used to measure magnetic fields up to

0.4 T. Time response of the sensor is 3 μs, average sensitivity under normal climatic conditions

is 2.4 mV/G.

Electrical scheme of the setup is presented in Fig. 2. Electrical energy is stored up in the

capacitor bank, with capacity 2850 μF and maximal voltage 5 kV, and then it is directed to the

coil through the mercury switch. The switch is closing when a triggering pulse is fed from the

G5- 56 pulse generator. Pulses from the magnetic field sensors are tested with the С1- 73

oscilloscope. Photographs of the pulses are registered by the camera and then processed on

PC. Because of great attenuation the first current pulse is only important. Value of the second

(reverse) pulse is no more than 17% of the first one. Next pulses are not taken into account.

Fig. 2.Scheme of the experimental setup. C- capacitor bank, L- magnetic coil, M- induction sensor, K- mercury

switch, PG- pulse generator, Ch- charger for capacitor bank.

Dependences of magnetic field on time are obtained in two ways. The first way to define B(t) is

to measure and process a signal from the Hall sensor (fields are less than 0.4 T). The second one

is to calculate the magnetic field by signals from the induction sensors using formula (3). An

expression for module of magnetic induction in the area of the sensor at t point of time is

There d is the diameter of the sensor winding, N= 10 is the number of turns, is the measured

induced emf.

V

Ch

PG

C

K

L

M

The normalized curves of magnetic induction in the center of coil are presented in Fig. 3. The

normalization is made to maximum magnetic induction. To make it easy the integration is

performed up to a peak of the second pulse. This makes additions in time and values of B up to

20 %.

Fig. 3.The curves of normalized magnetic induction in the center of coil during the first pulse. I1- induction sensor

with diameter 5.5 mm. M- Hall sensor. Voltages of charged capacitor bank are shown.

Peak values of the magnetic induction in the center and on face of the coil are given in table 1.

Values Bmax, T

Sensor 2,5 mm

Sensor 5,5 mm

Hall sensor

center, U0 = 10 V

0,018

0,017

0,034

center, U0 = 5 V

0,009

0,014

flank, U0 = 10 V

0,010

0,011

0,017

flank, U0 = 5 V

0,007

0,007

Table 1. Peak values of the magnetic inductions calculated using induction sensors and Hall sensor in the center and

on face of the coil along its axe. U0 - the voltage of charged capacitor bank.

One can see that the shapes of curves are in close agreement. The values of magnetic induction

calculated with (5) are 1-2 times less than these measured using the Hall sensor. The calculated

Bmax values are the same for both induction sensors within the error. The differences between

Bmax values calculated for the sensors in the coil center are more than these ones at the face.

The error in B values is connected with the field homogeneity and accuracy of the sensor

positions, besides the error of calculation. The dimensions of Hall sensor are 4×6 mm so it is

impossible to investigate the field homogeneity.

Thus, we can conclude that the induced emf in the sensors being used is directly proportional

to the rate of change of magnetic induction:

, and the difference in values of emf does

nor exceed a permissible error of the measurement. So, the emi law in form of (3) is applicable

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

0,00 100,00 200,00 300,00

I1, 10 V

M, 10 V

I1, 5 V

M, 5 V

t, μs

B/Bma x

to calculation of the induced emf in sensors placed axially inside the magnetic coil. It is in

agreement with the theory of induction coil sensors [14].

Measurement of induced emf in open-loop curvilinear conductors

Emf is induced invariably when magnetic power lines cross a conductor triggering changes of

the magnetic field. That was noted by M. Faraday in 1832. However, the law (3) describes the

only class of closed conductors. Emf induced in a curvilinear conductor may be caused by a

variable magnetic field or by movement of a conductor in the area of heterogeneous magnetic

field. A special case is the so- called unipolar induction, which is caused by movement of

electrons in a conductor moving in homogeneous magnetic field. Its mechanism is determined

by Lorentz forces acting upon the moving electrons.

Let us investigate the generation of induced emf in conductors of curvilinear form. Any spatial

curve with non- zero curvature may be approximated by a set of circlar arcs. Thus, the problem

of determination of induced emf in a curvilinear conductor is to find a value and direction of

induced emf in an arc. The parameters of arc are length, radius and position relative to

magnetic power lines.

Induced emf in a wire arc of finite curvature

The arcs are made of copper wire with 0.1 mm in diameter and are fixed on a dielectric cylinder

in the center of magnetic coil, normal to its axe (see Fig. 1). The ends of the arcs are bended so

that the current outlets are parallel to the axe (see Fig. 4).

Fig. 4. Shceme of a wire arc 1 fixed on a thin dielectric cylinder 2. L is a length, R=2.75 mm is a radius.

The curves of emf induced in the arcs with different length and fixed radius of 2.75 mm are

presented in Fig. 5. The time is limited by a peak of the second pulse, when derivative of B(t)

vanishes. Minimums of the curves correspond to the inflections of dependencies B(t) obtained

using Hall sensor.

R

L

B

1 2

Fig. 5. Induced EMF in the arcs (pieces of the turn). Length of a turn is 17,3 mm. The voltage of charged capacitor

bank is 100V.

One can see the curves differ by its stretch along the vertical axis and by slight shift along the

horizontal one. To compare emf let us take the minimum points. Relationships between the

emf absolute values in minimums and the arc length are presented in Fig. 6. Graph (a) is made

from the data given in Fig. 5, graph (b) from the average values of induced emf: for a whole

turn (2 measurings), half turn (4 measurings), 1/8 turn (3 measurings). In measuring, the arcs

are rotated in the plane normal to the coil axis. The relations may be approximated by straight

lines. Spread of points arises from inaccuracy of arc positions in the coil center and

heterogeneity of the magnetic field inside the coil. Graph (a) does not pass through the

coordinate origin because of the large spread of emf values in small arcs (up to 5 times for the

arcs 1/8 turn in length). Graph (b) is more exact, because it is obtained using average data.

From the physical point of view it is clear that induced emf is to be equal to zero in an arc of

zero length.

Fig. 6. Dependencies of absolute values of induced emf in arcs on their lengths. a - using data presented at fig. 5, b -

average values. The arcs are placed in the coil center, in the plane normal to the coil axis. Radius of the arcs is 2.75

mm.

-0,04

-0,02

0,00

0,02

0,04

0,06

0,08

0,00 100,00 200,00 300,00

1 round

1/2 round

1/4 round

1/8 round

t, μs

εi , V

0

5

10

15

20

25

30

35

0 10 20 30 40

l, mm

εi , mV

a

b

The family of curves of emf is obtained similarly for the arcs with different radii and of fixed

length 5.5 mm. But the resulting relationships are ambiguous. That is because of the value of

induced emf depends on the arc position in the normal plane. The emf decreases considerably

when the center of gravity of the arc is shifted to the coil center. The emf is lacking when a wire

segment is placed symmetrically in the center of coil. Power lines of the variable

magnetic field do not cross the segment. Small induced EMF is observed when the segment is

shifting from the center of the coil.

Thus, great spread of points of and differences in calculated values of magnetic induction

in the section above can be explained as follows. Shift of the inductive sensors in the normal

plane from the center causes low values of the magnetic induction. Shift of the arcs from the

center leads to increase of the induced emf. At that the summary magnetic induction changes

insignificantly, but its components in different pieces of the coil winding change essentially.

The proportionality of with fixed is in agreement with proportionality of to the area

of the sector of inscribed circle. However, must be proportional to the circular segment area

(the flux through other part of the loop is negligible). This contraries to proportionality to the

length and makes us to revise the law (3).

Induced EMF in a straight wire

The experimental setup (Fig. 2) is rebuilt to investigate the induced emf in a straight wire. A

ferrite rod is inserted into the magnetic coil. The rod is 9 mm in diameter, the length of its open

part is 140 mm. A segment of copper wire 400 mm long and 0.1 mm in diameter is placed

perpendicularly at a distance of 100 mm from the end of the rod (see Fig. 7). Minimal distance

from the wire to the rod is 2 mm. Voltage in the wire is registered by an oscilloscope.

Fig. 7. Scheme of measuring of induced emf in a straight wire. 1 is a segment of thin wire 400 mm long, 2 is a

magnetic coil, 3 is a ferrite rod, 4 is an oscilloscope.

The dependence of induced emf on time is shown in Fig. 8. The voltage of charged capacitor

bank is 100 V. One can see that the shape of the curve during the first pulse (200 μs) is similar

to the curves for windings and arcs. Some stretching of the curve and bumps after the pulse can

be explained by the ferrite retentivity.

1

2

3

4

Fig. 8. Induced EMF in a straight wire placed in the area of variable magnetic field near a ferrite rod.

The general law of EMI for curvilinear conductor

Two reasons of induced emf should be considered to formulate the general law of EMI for a

curvilinear conductor. The first one is the magnetic force acting at free charges in the conductor

that have average velocities close to zero. Presence of a variable magnetic field and asymmetry

of its distribution near a conductor (crossing by magnetic power lines) are necessary for origin

of this force. Let us suppose the direction of magnetic force is determined, by analogy with Lenz

force, by the vector product of relative velocity of the charge across magnetic power lines and

the vector of magnetic induction. Direction of a normal component in relation to the magnetic

power lines is important. This assumption is in agreement with obtained experimental data and

Lenz's rule. Direction of the velocity at any moment can be defined by a gradient and a sign of

derivative of the magnetic induction at the point of the charge. The magnetic force, acting at

the charge in some section of the conductor from the magnetic dipole placed at a spatial point

with coordinates x, y, z (see fig. 9), is:

Here q is the charge,

is the unit vector directed along the gradient of the magnetic induction,

is the magnetic induction produced by the dipole at the point of the charge location. Vector

is

collinear to vector

at every point of time.

-10

-5

0

5

10

15

20

0 100 200 300 400 500 600

εi , mV

t, μs

Fig. 9. Illustration of origin of magnetic forces acting at charge q in a conductor L by variable magnetic field of the

dipole. Directions of the forces are shown for increasing magnetic field, q < 0.

Summary magnetic force of all magnetic dipoles in the neighborhood V of the conductor is:

. Shift of the charges induces an electrical field. The shift is sufficiently lower

when small currents flow in the conductor with high concentration of free charges. Like

microscopic shifts causes stress in mechanics, the shift of charges causes electrical voltage.

Thus, in case of zero current is compensated by electrostatic force of the electrical field

induced in the conductor:

Let us obtain the required induced emf (voltage) integrating (7) along the line of conductor L:

As it follows from (8) the induced emf is proportional to the conductor length and to the sum of

derivatives of the magnetic induction of dipoles (their sets) at every point of the conductor

taking into account the orientation of the dipoles with respect to the conductor line.

Proportionality to the square in the law (3) means proportionality to the length and radius of

the turns. The proportionality to the length and derivatives in (8) is in agreement with (3). The

proportionality to the radius is evident from the following considerations. Asymmetry of effect

of the magnetic forces on the electrons in turns from the magnetic coil winding depends on

distances between the segments of the turns and the coil axis (or the winding). According to

Biot–Savart law, magnetic induction of the segment with current is inversely proportional to

the square distance from it. The sum of magnetic inductions (and their derivatives) of the

segments of the coil winding is proportional to the distance from the axis in sufficient

approximation. This dependence is provided for in the vector sum of magnetic forces in (8).

Verification of linearity of the dependence (squared dependence on the radius of a turn),

calculation of the induced emf in the turns of inductive sensors, and determination of the

proportionality constant in (8) are presented in the following section.

N

S

Bxyz Fм

x

y

z

0

exyz

q

L

Fэ

When a current is induced under variable magnetic field, the total voltage of induction

decreases due to the effect of self-induction. It should also be noted that the induced emf is the

demonstration of the phenomenon of EMI or with heterogeneities in the conductor structure

or the irregularity of influence of the magnetic forces along the circuit investigated. If the whole

circuit is placed in the area of a variable magnetic field and the potential drop is measured at its

points, the last one depends on a degree of uniformity of the field and distribution of resistivity

along the circuit. In case of total uniformity (for example, a closed loop with full symmetry of

magnetic field close to it) the potential drop is lack. The magnetic forces will only generate

induction currents in the turn. In this case the characteristic of the EMI effect is the magnitude

of inductive current multiplied by electric resistance.

Let us make an estimation of the magnitude of inductive current in a circuit with low voltages.

Let us use the provision of Drude's model, which describes the movement of electron gas in a

conductor [15]. When moving free electrons experience viscous friction from atoms of the

conductor. One can apply the second Newton's law to the current carriers, just as Maxwell's

method (1), but taking into account the viscous friction and expression obtained for summary

magnetic force of the variable magnetic field. So, the expression of movement of a current

carrier in the conductor is:

(9)

Here m is the mass of the current carrier, v(t) is its velocity at t point of time, α is the friction

coefficient (which is proportionate to conductivity of the conductor). Solving the differential

equation (9) in projection on tangent to the conductor for some heterogeneity function (the

magnetic force), we obtain the expression for v(t). Let us place it into the well- known

expression for current density in the conductor and get an estimation of the inductive current:

There n is concentration of the current carriers in the conductor.

The second reason of the EMI phenomenon is Lorentz forces, acting on moving charges in a

magnetic field. Unlike the magnetic forces described above, Lorentz forces do not move the

charges but curve their paths. In this case the phenomenon of EMI is observed in so- called

''unipolar'' generators. The first generator of unipolar induction was the disk of Faradey- Arago

[1], producing voltage of a few mV at great dimensions. However, the main mechanism of

generation of the induced emf in the disc is the mechanism described above. That is because

when the disk rotates the electrons experience an influence of the variable magnetic field (the

magnet is near the disk edge). One of the first unipolar generators of EMI only due to Lorentz

forces is the second experimental setup of Das Gupt [16]. Emf is generated in a conductive disk,

placed coaxial with a disk magnet, when the disk rotates singly or jointly with the magnet. The

induced emf in the unipolar generators is significantly less than the emf generated under

magnetic forces (6) in coils. However, the unipolar generators produce a high current that limits

their use in engineering [16]. Thus, the induced emf with mechanism of Lorentz forces makes a

small correction in the induced EMF for a curvilinear conductor. We have the expression for the

induced emf owing to Lorentz forces by analogy with derivation of (8):

Here v is the velocity of charges in the laboratory reference system, in the section of conductor

with coordinate l; B is the magnetic induction near the charge. Integration is performed along

the conductor line. Summary induced emf for a curvilinear conductor with no current is:

In the presence of current, the self- inducted emf should be considered, that reduces .When

the whole circuit is placed in the area of variable magnetic field, would be defined by the

similarity of the picture of magnetic field along the circuit.

Calculation of induced emf in windings using general law of EMI for

curvilinear conductor

Let us make a calculation to verify the law (8) and to establish quantitative correspondence

with the measured values of the induced emf in the induction sensors and wire turns. We

consider a sensor placed in the center of magnetic coil athwart to its axis (see Fig. 10). The

sensor winding consists of thin wire turns 1 mm in length that is significantly less than the

length of the coil winding (36 mm). Therefore, distinction of the magnetic field near the turns is not

significant. Thus, calculation of the induced emf may be done for one turn, placed in the center

of the coil. Multiplying the obtained emf by the number of the turns in the sensor, we have the

summary induced voltage, measured by the oscilloscope.

Fig. 10. Axial section of magnetic coil winding and induction sensor.

The emf in a turn of the sensor is induced by the variable magnetic field of currents flowing

along the turns of the magnetic coil. Let us number the turns n = 1, 2, 3, 4, counting off along

the axis from the centre; m = 1, 2, 3, 4, 5 is the number of a turn counted off athwart from the

R1

R2(m)

X(n,m)

ψ

n

m

1

2

3

4

5

1 2 3 4

axis. In view of symmetry of the task, we can consider only half of the turns, placed on one of

the sides of the plane of the sensor turn. We introduce the following notations: R1 is the radius

of the sensor turn; R2(m) is the radius of a turn of the coil with number m; x(m,n) is the

distance from the coil center to the center of the coil turn with numbers (m,n) on the axis; ψ is

the angle between the plane of the sensor turn and direction to the center of section of some

turn of the magnetic coil. Summing the emf from all turns of the coil, we obtain the emf in a

turn of the sensor.

The explanatory illustration for calculation of the induced emf in a turn of the sensor is given in

Fig. 11. Let us partition the source of magnetic field (the coil) into current elements (the sets of

magnetic dipoles inside them). The magnetic induction

of the field, produced by an element

of the current

, at the points of the sensor turn may is defined using Biot–Savart law. Let us

determine the magnitude and direction of the magnetic force , acting on electrons in the

sensor turn in the variable magnetic field of element dL, using formula (6) with the accuracy to

a constant factor. In view of symmetry of the task, the summary magnetic force, acting on the

electrons at some point of the sensor turn, is a doubled sum of the forces from a half - turn of

the coil. The required induced emf we obtain by projecting the summary magnetic force on the

direction of a tangent in a point of the turn in hand and integrating the projection along a

contour of the turn, in accordance with (8).

Fig. 11. Explanatory illustration for calculation of the induced emf

in a turn of sensor 1 from a turn of magnetic coil 2.

The calculations are carried out using MathCad 14 software. The expression for absolute value

of emf induced in a turn of the sensor by variable magnetic field of the coil is:

kR1 t( ) k 2 0tI t( )

d

d

1

5

m 1

4

n

R1 R2 m( )

0

'R1 m n( )

'R1 m n( )

sin R1 m n( )( ) sin R1 m n( )( )cos R1 m n( )( )cos R1 m n( )( )2

rR1 m( )2

R1 m( )

d

d

d

(13)

Here k is the desired proportionality factor in the law (8); μ0 is the universal magnetic constant;

I(t) is the time dependence of the current, flowing in a winding of the coil; α, β, γ, φ, θ, ψ are

the angles- functions (see Fig. 11); α' is the angle, at which the projection of the magnetic force

on the tangent to a turn of the sensor at the point with α=0 changes its sign; r is the projection

φ

B

e

F

r

R1

R2(m)

X(n,m)

dL I(t)

θ

γβ

ψ

E(t)

εi

1

2

on the plane of the sensor turn of the segment between element dL and the point of the turn

under consideration. The dependence I(t) is obtained by calculation of damped oscillations in

the circuit with magnetic coil (see Fig. 2), it takes the form:

I t( )

U0 e

Rt10 6

2 L

sin t10 6

L C

L

C

(14)

Here U0 is the voltage of charged capacitor bank, R=20 mOhm is the active resistance of the

circuit, L=1.9 μH is the inductivity of the coil, C= 2850 μF is the capacity of the capacitor bank, t

is the time in μsec. Inductivity and active resistance are selected so that the shape and time of

the dependence correspond to the experimental curves of the magnetic field inside the coil

B(t), obtained by Hall sensor. The curve I(t) is given in Fig. 12 for the first two pulses. Other

functions in (13) are explicit and determined using methods of stereometry. The angle α' is got

by finding of zero of the function γ(R1,α,m,n) by α.

Fig. 12. The calculated time dependence of current in magnetic coil.

The proportionality factor k in the expression (13) is determined by comparison of the

calculated values of the induced emf with five values measured using the induction sensor with

diameter 5.5 mm, and by introduction of a correction. Calculated curves of the emf induced in a

winding (10 turns) of the induction sensor are presented in Fig. 13 a, b. The curve of time

dependence (a) coincides with the experimental one within the measurement error. The curve

of the emf dependence on the turn radius (b) has a square- law shape to a first approximation,

that is in accordance with the law (3). The curve discontinuities are caused by imperfection of

the search algorithm for the roots of an equation in MathCad. However, shift of the minimum

of dependence (b) from the origin of coordinates points to imperfection of the calculation

model. One of the reasons of this problem are unaccounted slopes of the coil turns owing to

the continuous and irregular winding of the bus-bar. In this connection, a correction is

introduced in desired coefficient k that halves the previous value of k. The correction ensures

the agreement between the calculated and experimental values of the induced emf when

0100 200 300 400 500

100

0

100

200

300

A

300

100

I t( )

5000 t us

shifting the minimum of curve (b) to the origin of coordinates. Thus, the desired value of

proportionality factor in (8), (13) is: . The error is defined by spread of the

experimental values of the induced emf. Dimension of the coefficient is obtained by

comparison of the expressions for the induced emf (3) and (8).

Fig. 13. Calculated curves of the induced EMF in the induction sensor winding. a is the time-dependence when

radius of the winding is 2.75 mm; b is the dependence on radius of the winding at initial moment. The voltage of

charged capacitor bank is 10 V. The number of turns is 10.

A more exact experiment is needed to specify the proportionality factor in the law (8), with

different configurations of the magnetic field source and forms of conductors. Accuracy of

calculation of the induced emf will depend on the rate of partition of the field source into the

sets of dipoles (elements of the current) as well.

Conclusion

It has been ascertained that the well- known law of EMI (3) describes the induced emf in

conductive turns placed axially inside the magnetic coil. Measurements of the induced emf in

wire arcs with different lengths and radii have been carried out. Dependence of the induced

emf on the arc length proved to be linear that is in agreement with the law (3) applied to a turn.

However, this dependence does not fit the rigorous calculation of the induced emf in an arc

made by the law (3). Observed dependence of the induced emf on position of the magnetic field

source (coil winding) with regard to the conductor tells about need to take into account the symmetry of

influence of magnetic field on the conductor. The shape of the emf curve, induced in a straight wire,

is similar to this one for turns and arcs.

The general law of EMI for curvilinear conductor (8) is deduced from the obtained experimental

data. In deducing, the magnetic forces acting on a motionless charge () in a variable

magnetic field, as well as Lorentz force are considered. In a similar way the expressions for emf

(voltage) induced between points of a conductor of arbitrary form can be obtained. Calculation

of the induced emf in the sensor winding and comparison with the measured values are carried out to

verify the law (8) and to determine the proportionality factor. The obtained dependence of emf on a

radius is square- law to a first approximation that fits the law (3). The time dependence of emf

coincides with the experimental one within the measurement error. The general law requires

further testing for different configurations of the magnetic field relative to conductors.

0100 200 300 400 500

0.05

0

0.05

0.1

V

0.077

0.025

2.75 t( )

5000 t us

a

0 1 2 3 4 5 6

0

0.1

0.2

0.3

V

0.208

0.02

R1 0( )

60 R1 mm

b

When a current is induced in a conductor, it is necessary to take into account the decrease of

emf due to the self- induction effect and similarity of distribution of the magnetic field near the

conductor along the investigated circuit. The induced emf is absent when the distribution is similar

and heterogeneities in the circuit are absent. In this case the magnetic forces will synchronously

generate induction currents in every part of the conductor. Theoretical estimation of induction

current in a conductor with absence of voltages is given above.

Also it is necessary to note, the expressions for the induced emf are obtained under the

assumption of high concentration of free charges (elasticity of electron gas). If the

concentration is low (semiconductors, dielectrics) the induced emf will be lower significantly.

Thus, the induced emf in materials with low electrical conductivity, as well as in ionized

conductors should be investigated separately.

Discrimination of the forces acting on a quasi- stationary thermal charge (

) and on a moving

one (

) from magnetic field is relative in general. The nature of these forces is similar and is

connected with interaction of the induced magnetic moment of the charge with the magnetic

field. Absence of interaction between free electrons moving along magnetic power lines and

the field points to that own magnetic moment (spin) of the free electron is nill. Its magnetic

properties are demonstrated only when moving across the magnetic power lines. In this case,

the magnetic moment differs from dipole moment and has special character.

Understanding and application of the mechanism of electromagnetic induction may be of use in

electrical engineering. For example, the fact that the induced emf depends on position of a

source of magnetic field relative to a conductor can be used for development of special

induction sensors. It will enable measurement of magnetic induction, as well as determination

of spatial structure of the source. It will be of use in magnetography and magnetic tomography.

Acknowledgements

The author wishes to thank O. A. Petrenko, mechanical engineer of the Chair of General

Physics1, for his help in magnetic coils manufacture and Dr. M. V. Balabas for his valuable

comments on the work.

References

1. M. Faraday. 1947. Experimental investigation on electricity, v. 1. Moscow: ed. Academy

of Science of the USSR.

2. A. Saveliev. 1854. On works of academician Lenz on magnetoelectricity. Journal of

Ministry of Popular Enlightenment. 8 and 9.

3. Neumann F. E. 1849. Über ein allgemeines Princip der mathematischen Theorie

inducirter elektrischer Ströme. Berlin: Königl. Akad. d. Wiss.

4. J. C. Maxwell. 1865. A Dynamical Theory of the Electromagnetic Field. Phil. Trans. 155:

459-512.

5. J. C. Maxwell. 1954. Selected works on theory of electromagnetic field, pp. 56 - 88, 289 -

301, 644 - 645, 649 - 650, 661 - 663. Мoscow: GITTL.

6. Oliver Heaviside. 1950. Electromagnetic Theory, Vols. I, II, and III. New York: Dover.

7. R.F. Avramenko, L.P.Gratchev, V.I.Nikolaeva. 1976. Experimental verification of

differential laws of electromagnetic field. Scientific report. Мoscow: NIIRP.

8. Z.I. Doctorovich. 1995. Inconsistency of the electromagnetic theory and a way out of the

existing impasse. PMA N3.

9. I. E. Tamm. 2003. Principles of electricity theory. Moscow: Nauka.

10. E.M. Purcell. 1983. Electricity and Magnetism. Moscow: Nauka.

11. A. I. Korolev. 2013. Magnetodynamic waves in the air. JMMM. 327: 172–176.

12. A. Nussbaum. 1972. Faraday’s law paradoxes. Phys. Educ. 7: 231.

13. A. L´opez- Ramos, J. R. Men´endez and C. Piqu´e. 2008. Conditions for the validity of

Faraday’s law of induction and their experimental confirmation. Eur. J. Phys. 29: 1069–

1076.

14. S. Tumanski. 2007. Induction coil sensors—a review. Meas. Sci. Technol. 18: R31–R46.

15. N.Ashcroft, N.Mermin. 1979. Solid-state physics. Moscow: Mir.

16. L.A.Sukhanov, R.Kh.Safiullina, Yu.A.Bobkov. 1964. Electric unipolar machines. Moscow:

VNIIEM.