_{1}

The motional EMF in segments of the copper wire with magnetic shielding is found according to the voltage lack in two coils with partial shielding moving relative to the magnetic field lines. The first coil moved across the Earth’s magnetic field lines. The second one was located near the end face of the rotating disk magnet with an axial magnetization. Permalloy foil wound around a part of turns containing w ire was used for shielding. The experiments result reveals the penetrating ability of the magnetic field through the ferromagnetic shield and shows the physical nature of the superposition principle. With this in mind, a universal method for calculating EMF induced in a conductive body has been provided as well as the concepts of magnetic field lines velocity and acceleration have been introduced.

In magnetostatics, it is believed that with a passive magnetic shielding the magnetic field flow is attracted into the magnetic shield area. Magnetic field practically does not penetrate [

_{x} and vertical B_{y} components. Magnitude of the horizontal intensity of the Earth’s magnetic field at the place of observation (St. Petersburg) is H g ≃ 15 µT according to [_{x} equals to 20.4. Thus, the magnitude of B_{x} can be calculated according to the formula (1):

B x = μ H g cos ( 20.4 ˚ ) ≅ 14 μT (1)

Here μ ≃ 1 is the magnetic permeability of the air atmosphere.

The experiment was carried out inside a moving vehicle. The weakening of the magnetic field by the vehicle’s steel body at the place of measuring (near the window) was not significant, since the compass with a magnetic arrow operated like outside the vehicle’s body as well.

In coil sides, the motional EMF caused by the Lorentz forces acting on the free electrons of the conductor appears. Lorentz force F_{Ly} acting along the upper and lower sides of the loop are mutually compensated. Lorentz forces along the sides F_{Lx}, according to the existing theory, are not to be compensated due to the

shielding of the magnetic field on the side with permalloy shield. The shielding effectiveness of the thin-walled cylinder from N layers of foil can be evaluated [

T = D μ N d (2)

Here T is the coefficient of magnetic field attenuation inside the cylinder, μ ≥ 50000 is the coefficient of magnetic permeability of permalloy, D is the cylinder diameter, d is the foil thickness. Thus, T ≤ 0.02 . Taking into account the edge effects, the attenuation coefficient should be slightly higher. The amount of EMF in the coil sides that should be indicated by voltmeter can be calculated according to the formula (3):

ε i Σ = ( 1 − T ) ⋅ v ⋅ B x ⋅ l ⋅ n (3)

Here l is the width of shielding foil, n is the number of wire turns. Under the conditions described, a voltage indication can be expected on the voltmeter ε i Σ ≃ 47 mV .

The magnet is made of NdFeB with the amount of retained magnetization, according to the description, B r = 1.44 - 1.48 T . The value of residual magnetization was determined after measuring the magnetic field near the magnet end face using the Hall sensor. The calculation according to [

B r = 2 B z ( D + z R 2 + ( D + z ) 2 − z R 2 + z 2 ) − 1 (4)

Here B_{z} is the magnitude of magnetic induction on the magnet axis at a distance from the end face z. The measured value at z = 20 mm was 908 Gs, which corresponds to the magnitude of B r ≃ 1.27 T . The magnet is revolved by motor at angular velocity ω ≃ 425 rad / s . To determine the angular velocity, the photographing of rotating magnet with two marker nuts secured diametrically opposite on the magnet sidewall was used. With the ambient lighting, the nuts leave shiny tracks, see photo on

The textolite disk 6 of radius R* = 20 mm and thickness of 15 mm is brought to the rotating magnet from below. The coil 7 consisting of n = 10 copper wire turns with cross-section of 0.03 mm^{2} in fluoroplastic insulation is secured to the disc. The external part of the coil is located along the disc circumference, the internal―along its diameter. One radial part of the coil is placed in the shielding cavity 8. The cavity is constructed in the form of cylinder 5 mm in diameter containing 15 layers of permalloy foil with a thickness of 20 μm. The cylinder is unevenly flattened due to its securing to the disc with the help of Scotch tape. The maximum ratio of the ellipses’ semiaxes lengths in the resulting cavity

section reaches two. The evaluation of the magnetic field attenuation coefficient inside the cavity according to the formula (2) gives a value T ≃ 0.00033 . For ease of use, the textolite disc is provided with a handle in the form of wooden rod 9. The coil terminals are arranged along the magnet axis. The coil voltage is measured by multimeter in voltmeter mode with accuracy of 1 mV.

When approaching the coil to rotating magnet from below, the permalloy shield is magnetized, and the attractive force appears. This force destabilizes the magnet rotation. To reduce the magnet beating in the vertical direction, it was slightly directed by the side hand touch. With this, rotation speed reduced and the reduction magnitude could be measured by changing the tone of the engine sound. By sound, this reduction did not exceed two times, that is ω ≥ 200 rad/s. With the distance between the coil diametric part and the magnet surface of the 10 mm the beats did not exceed ±2 mm. In this case, estimation of the magnetic field magnitude in the area of the coil diametric part made according to the formula (4) gives a value of B ≃ 0.16 T T. When rotating the magnet, in the coil two radial Lorentz forces appears: F_{L}_{1} и F_{L}_{2}, one of which, due to magnetic shielding, might be close to zero. The amount of the motional EMF induced in the radial parts of the coil can be estimated by the formula (5):

ε i Σ = ( 1 − T ) ⋅ v ¯ ⋅ B ⋅ R ∗ ⋅ n (5)

Here v ¯ = 1 2 ω R * is the mean linear velocity of the relative motion of the coil radial parts and the magnetic field lines. The voltage indication can be expected on the voltmeter ε i Σ ≥ 64 mV .

The result of the voltage measurements in both experiments is the same. The measured value is 0 ± 1 mV. Possible sources of the measurement errors are: Heterogeneity of the magnetic field, the edge effects near the end faces of the shielding cavities, the inaccuracy of the coils orientation relative to magnetic field lines, gaps between the layers of permalloy foil, real value of the permalloy magnetic permeability coefficient different from the reference one. These all can not impact significantly during the measurement of voltages with a precision of 2% for experiment No.1 and 1.5% for experiment No.2.

The result obtained is explained with the presence of the motional EMF in the conductors located inside the shielded cavities. With this, the magnitude of this EMF coincides accurately with those one in the unshielded sections of the conductors. This indicates the penetrating ability of the magnetic field through the ferromagnetic shield. Thus, the superposition principle takes the physical meaning. The magnetic field at a point is a result of physical superposition of the elementary magnetic fields from individual dipoles including the induced ones. Such representation of the superposition principle is, for example, here [

Work [

If remember the classical picture of magnetic field lines as the streamlines of luminiferous medium, one can give the universal method for calculating the electromagnetic induction that occurs in a conductive body. The correctness of such approach is confirmed, for example, in [

F m = q ∑ i = 1 n [ v i , B i ] (6)

Here B_{i} is a vector of magnetic induction generated at a charge location point by the elementary field source with number i. v_{i} is velocity vector of the charge movement relative to the field lines of i source. For the sake of practical calculations, the amount can be replaced with a spatial integral in terms of the source volume provided that its size ≫ of the elementary magnetic dipoles. v_{i} can be represented as amount (7):

v i = v i τ + v i r (7)

Here v_{iτ} is a vector of peripheral component of velocity that is tangential to surface with B i = C o n s t . v_{ir} is a vector of radial component that is directed perpendicular to the surface. The magnitude of the latter can be found in the formula (8):

v i r = 1 | ∇ B i | ⋅ ∂ B i ∂ t (8)

The expression for induced EMF in [_{m} having effect on a single charge along some path L between the measurement points of potential (9).

ε i n d = − 1 q ∫ L ( F m , d l ) (9)

The independence of ε_{ind} on the integration path is true provided that there is a homogeneous magnetic field moving at a fixed speed in the conductive body under study. If this is not the case, i.e. the pictures of the charges movement relative to the field lines in various paths are different, it is necessary to break the body into small segments, calculate the force in each segment according to the formula (6) then make integration according to the formula (9) by all possible paths of current. Then the resulted array of ε_{ind} is added according to a known addition formula of EMF sources (paths) subject to their electrical resistance. The calculation of ε_{ind} becomes much simpler when considering the thin conductor segment. In this case, the integration is carried out by its spatial curve.

During the charge movement in the homogeneous magnetic field, the addition of the force tangential components F_{m} will result in the Lorentz force. With this, the radial components are mutually compensated. Further integration will provide the clear motional EMF.

The charges movement in the inhomogeneous alternating magnetic field across the magnetic field lines is implemented, for example, in the standard problem when placing a wire turn inside the long solenoid, symmetrically to the axis of the latter. Induced EMF in the turn is caused by the alternating magnetic field of the solenoid and is calculated according to the faraday’s law of electromagnetic induction. In the law, the flux rate change of the magnetic induction passing through the turn corresponds to the expression (9) with the substitution (8).

v_{ir} can be expressed via derivative of magnetic induction (8). v_{iτ} is not expressed via B_{i}. This points to the necessity of introducing such a parameter of the magnetic field as the displacement velocity of the magnetic field lines v_{B}. In a static case, it will be zero. In a dynamic case, the three-dimensional picture of v_{B} distribution together with a picture of magnetic field lines will give a more complete description of the magnetic field in some area. Due to the fact that the magnetic field has a finite rate of propagation, it is better to call it “magnetic flow” when considering fast processes. Streamlines slipping and rotation of these flows can be characterized by the distribution of v_{B} velocity and, if necessary, by acceleration of a_{B} at any time. At periodic movements it can be said about the expansion of “magnetodynamic waves” [

During two experiments, the presence of motional EMF inside the ferromagnetic shields made of permalloy foil has been detected. Magnetic coils with partially shielding windings and placed in homogeneous magnetic fields were used. During the first experiment, the coil moved relative to the Earth’s magnetic field, and during the second one, the magnetic field lines of the rotating disk magnet moved relative to the coil. The lack of voltage on the coils terminals indicates with a good accuracy the presence of motional EMF in the shielded parts of windings equal in magnitude to those ones in parts without a shield. This reveals the penetrating ability of the magnetic field through the ferromagnetic shield and shows the physical nature of the superposition principle. As a result, the universal method for calculating the induced EMF that occurs in the conductive body moving relative to magnetic field was represented. It withdraw the paradoxes in calculating induced EMF on the classical law [

Awareness of motional EMF presence inside the shielded cavity can be used in the technique of, for example, the shielded signal cables laying in fast-moving vehicles. The nonequivalence of the rotating magnetic field to the stationary one with the fixed picture of the field lines is the cause of the EMF emerges in the external circuit of unipolar generator when the disk rotates along with the magnet [

Korolev, A.I. (2018) The Motional EMF in Conductors with Ferromagnetic Shield. Open Access Library Journal, 5: e4461. https://doi.org/10.4236/oalib.1104461