ELECTRICAL ENGINEERING Displacement current and Ampère's circuital law Ivan S. Bozev, Radoslav B. Borisov

The existing literature about displacement current, although it is clearly defined, there are not enough publications clarifying its nature. Usually it is assumed that the electrical current is three types: conduction current, convection current and displacement current. In the first two cases we have directed movement of electrical charges, while in the third case we have time varying electric field. Most often for the displacement current is talking in capacitors. Taking account that charge carriers (electrons and charged particles occupy the negligible space in the surrounding them space, they can be regarded only as exciters of the displacement current that current fills all space and is superposition of the currents of the individual moving charges. For this purpose in the article analyzes the current configuration of lines in space around a moving charge. An analysis of the relationship between the excited magnetic field around the charge and the displacement current is made. It is shown excited magnetic flux density and excited the displacement current are linked by Ampere’s circuital law. ъ а аа я аъ а ъя (Ива . Бв, аав Б. Бв.) В я, я , я , яя . , я : , я я. я я, я я . я . К , я ( я я , я, я я я. З я я я. я я. , я я я я .

1. Introduction configurations of the electric field of moving charge are shown on Fig. 1 and Fig. 2. First figure represents The size of electronic components constantly delayed potentials of the electric field according to shrinks and the discrete nature of the matter is Liénard–Wiechert and this picture is not symmetrical becomming more obvious. Electromagnetic field towards the axis of the moving charge. In this case the surrounding individual charges cannot be neglected =v c anymore and the processes connected to the picture represents charge moving at speed 0,5 0 , electromagnetic field energy should be considered. where c0 is the speed of light in vacuum. Doppler Most famous analysis discussing the electromagnetic effect is expressed and presented on the figure. Figure field surrounding moving charges are Liénard– 2 represents electric field strength of the electrical Wiechert (Liénard–Wiechert potentials), Oliver field according to the analysis of Heaviside and as it is Heaviside and Oleg Jefimenko (Jefimenko's shown, the picture is symmetrical about the moving equations). All of the above analysis is so called charge. The compression along the x axis can be solutions of the Maxwell equations. Most famous explained using Lorentz contraction in the direction of

“Е+Е”, 11-12/2016 15 the movement. Later on Efimenko introduces the where ε0 is electrical constant in vacuum, Е is the concept „delayed time“. electric filed strenght and D is the electric flux density. y q v Figure 3 shows moving charge at speed along x the axis. Electric field strenght Е and the electric flux density D are defined according to the Coulomb V law for point charge, where it matches the border case where v → 0 . v x y q x r

y

q Fig. 1 vt x v

Fig. 3

y From the equations below we can see that they have spherical symmetry qrˆ qrˆ E D E (2) = 2 , ε= 0 = 2 , 4πε0r 4πr E 2 2 2 2 where r = x + y + z when we have the three axis q v x coordinate system and rˆ is the unit vector of r . Figure 3 represents two axis coordinate system 2 2 2 ( r = x + y ; z = 0) and the symmetry is along the x axis. When the charge is moving, the electric field strength for the individual points in the space is function of time t. According to equations (2) the Fig. 2 electric field strength Е is:

According to all analysis of the electromagnetic qrˆ qrˆ (3) E = , E = . field surrounding moving charge causal connection x vt 2 y 2 z 2 x vt 2 y2 4πε0 ()− + + 4πε0 ()()− + between the separate quantities and processes is () missing. Respectively the three components of the 2. Exposition derivatives of electric field strength are decomposed as follow: We know from Maxwell equations [1], that the j displacement current density D is: ()x − vt ()xq − vt (4) x E=E ⋅ = , 1 3 D E 2 2 2 2 2 2 ∂ ∂ x − vt z+y+ 2 4πε0 x − vt z+y+ 2 (1) jD = ε= 0 , ()() ()() ∂t ∂t y qy (5) E y E= ⋅ = , 1 3 x vt 2 2 z+y+ 2 x vt 2 2 z+y+ 2 ()()− 2 4πε0 ()()− 2

16 “Е+Е”, 11-12/2016 z qz (6) z E=E ⋅ = . 1 3 y x vt 2 2 z+y+ 2 x vt 2 2 z+y+ 2 ()()− 2 4πε0 ()()− 2

For the first derivatives with respect to time for each axis we can write: j 2 2 2 2 2 2 qv 2()x − vt − y − z qv 2x − y − z (6) E′x = ⋅ ( ) , Ex′ = ⋅ ( ) , x 4 5 4 5 v πε0 2 2 2 t→0 πε0 2 2 2 ()()x − vt + y + z 2 ()x + y + z 2 j qv 3 ()xy − vt qv 3yx (7) E′y = ⋅ , E′y = ⋅ , 4 5 4 5 πε0 2 2 2 t→0 πε0 2 2 2 ()()x − vt + y + z 2 ()x + y + z 2 qv 3 xz vt qv 3zx E ()− E (9) ′z = ⋅ 5 , z′ = ⋅ 5 , 4πε0 2 2 2 t→0 4πε0 2 2 2 ()()x − vt + y + z 2 ()x + y + z 2 4.Fig Accordingly for the displacement current density jD the following equations are valid: qv 2x2 y 2 z 2 j E ()− − (10) x ε= 0 x′ = ⋅ 5 , 4π 2 2 2 ()x + y + z 2 qv 3yx j E (11) y ε= 0 ′y = ⋅ 5 , 4π 2 2 2 ()x + y + z 2 qv 3zx j E (12) z ε= 0 z′ = ⋅ 5 . 4π 2 2 2 ()x + y + z 2 Selection of the analyzed point to be in the plane of xy simplifies the analysis (follows that z=0) and the distribution of the displacement current can be represented clearly. For the analysis were used online Fig.5. calculators [2] and [3]. Use of the above mentioned calculators give advantage that they provide results for Fig. 6 presents the shape of the displacement each step of the transformations and function’s visual current and excited magnetic induction B around a representation using graphics. moving charge. The value of the magnetic induction A simulation has been conducted with respect to according to Fig. 6 is calculated using Biot-Savart the displacement current using software [4] and Fig 5 law: shows the results from the calculations of equations vq rˆ v sinθ (10) and (11) in case z=0. B µ0 × µ0 (13) = 2 = 2 . Displacement current trajectories shown on Fig. 4 4π r 4π r in the xy plane are approximate. Paths flow of the In principal it is interesting to be determined if in displacement current are represented by oval curves, the space surrounding a moving charge it`s magnetic close to circles, symmetrical with respect to y axis and field and displacement current are connected through passing through the origin of the coordinate system (in the Ampère's circuital law. this case through the moving point charge). In case of Calculations were conducted for the covered uniform motion of the charge q and speed tend to be current by the contours in shape of circle with center zero ( v 0 ), the vector field of the displacement → on x axis and magnetic flux density along the circle is current is solenoidal. It has divergence equal to zero checked if with Biot-Savart law satisfies the Ampère's and the displacement current forms closed curves in circuital law. Table 1 represents the results from the the form of pipes where the currents are constant. calculations in case q = 1 C, r = 1 m and v =1 m/s . First column represents the angle θ.

“Е+Е”, 11-12/2016 17 The results for calculated magnetic flux density If θ = 30° the equation becomes { were done using calculator [5], where equation (13) y((2)(cos(pi/6))^2-y^2)/(2((cos(pi/6))^2+y^2)^(5/2)) } was used and previously mentioned values using the online calculator [2]. 7 ( B =10− sinθ ). Second column contains the In case θ = 90° the circle covers the beginning of calculated results. the coordinate system where the values are infinite Third column contains the full current according to and the integral becomes divergent. Approximate the defined magnetic flux density on the result could be derived in case we select a coordinate system very close to the beginning of the initial y coordinate system and integral is defined from 0 to 1. In case x = 0 the limits of the integral become y =1...∞ and definite result is obtained with negative B sign. This is possible, because the vector field of the j current is solenoidal (no divergence) and the currents r through the two half’s of the plane, separated by the θ circle with centre in the beginning of the coordinate system are equal and with opposite signs. x Results from the calculations are equal to the one from vq the previous column. B The fourth column contains the current, defined by spherical surface surrounded by the circle according to B angle θ, where the expressions j zx 0 and j zy 0 are B ()= ()= j used. The current follows through the equation:

1 i j S j S 2 ydx cd = ()x ⋅ x⊥ + y ⋅ y⊥ π = cos θ 6.Fig (17) 2 2 x 2x − y + 3yx R qv ()y 1 x circumference of the sphere defined by the angle θ. ydx .dx = 5 2π = 3 cos 0 4π 2 2 2 θ r ()x + y B 1 µ0 v sinθ 1 (14) i = 2πrH = 2πr = 2πr = sinθ . r 2 r µ0 µ0 4π 2 Equation (17) becomes { x/(1^3) } in case θ = 30° with limits (cos30°... .1) and again we use online Fourth column represents the current passing calculator [2] and the results are the same. through the circle and covered by the circle, defined Fifth column defines the current using calculator by the angle θ. [6] for spherical coordinates. The surface integral If z = 0 equation (10) can be written as: doesn’t exist in the calculator, so a volume integral in a very thin layer on the surface of the sphere is 2 2 qv ()2x − y calculated. Following equation is used for the (15) jx ε= 0 Ex′ = ⋅ 5 . 4 2 2 displacement current component j perpendicular to π x + y 2 ρ () the sphere surface: If we take x rcosθ , y rsinθ and r x2 y 2 , qv cos = = = + j θ (18) ρ = 3 . then for the current passing through the circle the 2π ρ following equation is valid: Current through the surface of the sphere is limited 2 sqrt ()1−x qv 2x2 y 2 by the angle θ and is given by the integral in spheric i ()− yy p = 5 2π d = coordinates, where [J] is jacobian for spherical (16) 0 4π 2 2 2 ()x + y coordinates ([J]=ρ2sinθ) and ρ is the radius of the 2 2 sinθ y 2 cos y ()⋅ ()θ − .dy sphere. It is volume integral, because this is the only 5 0 2 2 calculator found by the authors online and accessible. 2 cosθ + y 2 ()() In order to get surface integral very thin layer on the surface of the sphere was taken.

18 “Е+Е”, 11-12/2016 iS J j d dθρ d the mathematical model used to represent the ⊥ρ = []ρ ϕ = (19) V()S electromagnetic field, proposed in series of articles, cos 2πρθ 2 2 sin θd d .d citated in [7]. = 00ρ1 ()ρ θ 3 ρ θ ϕ 2πρ Interesting research in future will be the connection between the magnetic flux density in Expression used for the calculations in [6] is particular point of the space around moving particle as {(10^10)(sinθ)*cosθ/(2*pi*ρ)} and limits ρ1=1; function of volume integral in whole space. The ρ2=1.0000000001. function under the integral is vector multiplication of Table 1 the displacement current density at the particular point and the distance from the current point to particular 1 2 3 4 5 6 point in the space.

θ B i ip icd ic This paper was originally published in the XXIV deg T (х10-7) A A A A National conference with international participation TELECOM, Sofia, Bulgaria, 2016. 30 0.5 0.125 0.125 0.125 0.125 45 0.707106 0.25 0.25 0.25 0.25 REFERENCES [1] IEC 60050, International Electrotechnical 60 0.866025 0.375 0.375 0.375 0.375 Vocabulary (IEV): Area: 121: Electromagnetism, Section 90 1 0.5 0.5 0.5 0.5 121-11: Electromagnetic concepts and quantities. http://www.electropedia.org/ Т [2] Integral Calculator, From the results in Table 1 we can conclude http://www.integral-calculator.com/ following: 1. Excited magnetic flux density and displacement [3] Derivative Calculator http://www.derivative-calculator.net/ current around moving charge satisfy the Ampère's circuital law. [4] Scilab 5.4.1 (64 –bit) - Open source platform for numerical computation. 2. Vector field of the displacement current in case of constant speed of the moving charge is [5] Scientific Calculator - Web2.0calc, solenoidal field. http://web2.0calc.com/ 3. If we consider that the electrical charge exists in [6] Triple integrals in spherical coordinates calculator, discrete form and is concentrated is tiny volume http://www.wolframalpha.com/widgets/view.jsp?id=89c969 of the space they could be considered only as c21b169fa996f899d9b2a9858 exciters of the induced electrical current and the [7] Bozev, I., R. Borisov. Experimental study on magnetic field is excited by the displacement electromagnetic induction of AC in areas with zero magnetic flux density. XXIII Telecom 2015, NSTC, Sofia, current. Bulgaria, 2015. (In Bulgarian). 4. In practice it is much more convenient if we use conduction current and convection current, which are much more clear and are Assoc. Prof. Dr. Ivan Stefanov Bozev mathematically easier to present, while the tel.: 0888461118 -mail: [email protected] results are the same. Eng. Radoslav Borislavov Borisov 3. Conclusion Department of Microelectronics, Technical University This article presents theoretical analysis and of Sofia, Kl. Ohridski, Blvd., 1756 Sofia, calculations showing the connection between the e-mail: [email protected] magnetic flux density in space surrounding moving charge and the displacement current through the Received on: 23.11.2016 Ampère's circuital law. Derived connections improve

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