Exercises in Electromagnetic Field

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1 DR. GYURCSEK ISTVÁN Exercises in Electromagnetic Field Sources and additional materials (recommended) Gyurcsek I. Elmer Gy.: Theories in Electric Circuits, Globe Edit 206, ISBN: Simonyi K.: Villamosságtan. AK Budapest 983, ISBN:

2 Stationary Electric Field EMF.0 Find I electric current, flows through the aluminum rod when mv is connected to its terminals. The length of rod is 0 cm, diameter is 5 mm and Al 0,025 mm 2 /m. Surfaces, where current enters / leaves the aluminum body, supposed to be equipotential. Solution diff. Ohm s law J σ E ρ E l I න A J da ρ න A E da V න E dr E l E V l 0 A I ρ V l A V ρ l V R A A d2 π 4 25 π mm 2 R ρ l A μω I V R A 2 gyurcsek.istvan@mik.pte.hu

3 Stationary Electric Field EMF.02 Find I AB then I CD when mv is connected to AB then CD terminals. Al 0,025 mm 2 /m Surfaces, where the current enters and leaves the aluminum body, supposed to be equipotential. Solution R ρ l A l σ A Integral formula with conditions Homogeneous material or is constant Constant length beside cross-section Constant cross-section beside length 3 gyurcsek.istvan@mik.pte.hu

4 Stationary Electric Field R ABm ρ l A R CDm ρ l A 4 gyurcsek.istvan@mik.pte.hu

5 Stationary Electric Field A m h d m π 2 h d k + d b 2 2 π 020 mm 2 l AB d k d b 2 70 mm 0.07 m R ABm ρ l AB A m Ω I ABm V R ABm ka 5 gyurcsek.istvan@mik.pte.hu

6 Stationary Electric Field l m d m π 2 d k + d b 2 2 π mm m A CD h d k d b mm 2 R CDm ρ l m A CD Ω I CDm U R CDm 685 A 6 gyurcsek.istvan@mik.pte.hu

7 Stationary Electric Field dr AB ρ r k R AB න dr AB න r b l A(r) ρ dr h r π r k ρ r b dr h r π ρ h π න r b r k r dr 7 gyurcsek.istvan@mik.pte.hu

8 Stationary Electric Field R AB ρ h π න r b r k r dr ρ h π ln r r b r k ρ h π ln r k ln r b ρ h π ln r k r b R AB π ln Ω I AB V R AB 5,29 A 5,834 A 8 gyurcsek.istvan@mik.pte.hu

9 Stationary Electric Field dg CD dr CD r k G CD න dg CD න rb r k r b da ρ l(r) h dr ρ r π h dr ρ r π h ρ π න rk rb r dr 9 gyurcsek.istvan@mik.pte.hu

10 Stationary Electric Field G CD h ρ π න rk rb r dr h ρ π ln r r k r b h ρ π ln r k ln r b h ρ π ln r k r b G CD π ln S I CD V R CD R CD G CD Ω A 685 A 0 gyurcsek.istvan@mik.pte.hu

11 Stationary Electric Field EMF.03 Calculate the earth resistance when the ground is wet and when it is dry

12 Stationary Electric Field dr l σa(r) dr σ 2π r 2 R f σ 2π r rg σ 2π r G 2π σ r G R f න r G dr σ 2π r 2 σ 2π න r G r 2 dr R fwet 2π σ WET r G 3.83 Ω R fdry 2π σ DRY r G 38.3 Ω 2 gyurcsek.istvan@mik.pte.hu

13 Stationary Electric Field EMF.04 Find the electric potential of the rod when the ground is wet and when it is dry. V fwet I R fwet kv V fdry I R fdry kv 3 gyurcsek.istvan@mik.pte.hu

14 Stationary Electric Field EMF.05 Calculate the step voltage between the m and.5 m distant points to the lightning rod when the ground is wet and when it is dry. 4 gyurcsek.istvan@mik.pte.hu

15 Stationary Electric Field R,2WET 2π σ WET r r 2 2π Ω R,2DRY 2π σ DRY r r 2 R,2 r 2 2π σ න r 2 dr 2π σ r 2 r r r 2π σ r r 2 2π Ω V,2WET I R,2WET V V,2DRY I R,2DRY kv 5 gyurcsek.istvan@mik.pte.hu

16 Stationary Electric Field Another way... J r I A r I 2π r 2 E r σ J(r) I σ 2π r 2 V,2 න r r 2 E(r)dr න r r 2 I σ 2π r 2 dr I 2π σ න r r 2 r 2 dr I 2π σ r r 2 I V(r) 2π σ r 6 gyurcsek.istvan@mik.pte.hu

17 Stationary Electric Field EMF.06 Calculate the dissipated power in earth when the ground is wet and when it is dry. P WET V fwet I MW P DRY V fdry I MW 7 gyurcsek.istvan@mik.pte.hu

18 Stationary Electric Field Another way (differential Joule s-law) p r E r J r P V න V P V න r G P WET p r dv, I σ 2π r 2 I 2 σ 4π 2 r 4 2π r2 dr I 2π r 2 I 2 σ 4π 2 r 4 dv A dr 2π r 2 dr I2 2π σ න r G π MW P DRY r 2 dr I 2 2π σ r G π MW 8 gyurcsek.istvan@mik.pte.hu

19 Stationary vs. Static EF EMF.07 Calculate the electric current through each capacitor if the connected voltage is V 2 V. Find the dissipated power in each of layers and within the capacitors. Parameters A 20 cm 2 d 2,5 mm / m / m d mm d 2,5 mm A 8 cm 2 A 2 2 cm 2. 9 gyurcsek.istvan@mik.pte.hu

Stationary vs. Static EF R a d σ A + d 2 σ 2 A I a V R a R ρ l A l σ A 0 3 6 0 8 20 0 4 +.5 0 3 3 0 9 8.33 + 250 258.33 MΩ 20 0 4 2 258.33 06 46.45 na P a I a 2 R a 46.45 0 9 2 8.33 0 6 7.

20 Stationary vs. Static EF R a d σ A + d 2 σ 2 A I a V R a R ρ l A l σ A MΩ na P a I a 2 R a nw P a P a + P a nw Comment Notice the analog behavior of static and stationary electric field in series capacitors. C a Q V C a + C a2 G a I V G a + G a2 P a2 I a 2 R a nw A + ε d A + σ d ε 2 A d 2 σ 2 A d 2 20 gyurcsek.istvan@mik.pte.hu d ε A + d 2 ε 2 A d σ A + d 2 σ 2 A ε ε 2 A ε 2 d + ε d 2 σ σ 2 A σ 2 d + σ d 2

21 Stationary vs. Static EF R b R b R b2 I b V R b d σ A μa d σ 2 A MΩ P b P b + P b2 V2 R b + V2 R b μw Comment Notice the analog behavior of static and stationary electric field in parallel capacitors. A C b ε d + ε A 2 2 d A G b σ d + σ A 2 2 d 2 gyurcsek.istvan@mik.pte.hu

22 Stationary vs. Static EF EMF.08 Find the leakage current through insulating layers in the km long high voltage coaxial cable and also calculate the dissipated power in the insulation. r 0,5 cm r 2,5 cm r 3 2 cm r 4 2,2 cm 0-8 / m / m V 7 kv 22 gyurcsek.istvan@mik.pte.hu

23 Stationary vs. Static EF R σ l 2π න r r2 r dr + σ 2 l 2π න r2 r3 r dr R l 2π σ r2 r3 ln rቚ + ln rቚ r σ 2 r2 dr dr σ(r) A(r) dr σ(r) l 2π r R 2π l ln r 2 r σ + ln r 3 r2 σ 2 r3 R න r dr r2 σ(r) l 2π r න r dr r3 σ l 2π r + න r2 dr σ 2 l 2π r 2π ln ln kω 23 gyurcsek.istvan@mik.pte.hu

24 Stationary vs. Static EF I V R ma, P V I W Comment Notice the analog behavior of static and stationary electric field in bilateral capacitor & cable. C 2π l ln r 2 ln r 3 r r2 ε + ε 2, G 2π l ln r 2 r σ + ln r 3 r2 σ 2 24 gyurcsek.istvan@mik.pte.hu

25 Stationary vs. Static EF EMF.09 Find the leakage current through insulating layers in the spherical capacitor and also calculate the dissipated power in the insulation. r 0,5 cm r 2,5 cm r 3 2 cm r 4 2,2 cm 0-8 / m / m V 7 kv 25 gyurcsek.istvan@mik.pte.hu

26 Stationary vs. Static EF r3 R න r dr r2 dr σ(r) 4π r 2 න σ 4π r 2 + න σ 2 4π r 2 r r2 r3 dr R r2 σ 4π න r 2 dr + r3 σ 2 4π න r 2 dr r r2 dr dr σ(r) A(r) dr σ(r) 4π r 2 R 4π σ r อ r2 r r3 + อ σ 2 r r2 R 4π r r 2 σ + r 2 r 3 σ 2 4π MΩ 26 gyurcsek.istvan@mik.pte.hu

27 Stationary vs. Static EF I V R ma, P V I W Comment Notice the analog behavior of static and stationary electric field in spherical capacitor & conductor. C 4π r 2 r r r 2 ε + r 3 r 2 r 2 r 3 ε 2, G 4π r 2 r r r 2 σ + r 3 r 2 r 2 r 3 σ 2 27 gyurcsek.istvan@mik.pte.hu

28 Temp Dependency EMF.0 The electric current through a 230V/00W electric bulb, connected to the main voltage at 20 C ambient temperature is times higher than the current at 2400 C operating temperature. Find the temperature coefficient of tungsten filament. R n R 0 + α θ n θ 0 P n 00 W I 0 I n V R0 α θ n θ 0 0 α 0 θ n θ 0 R n + α θ V R n θ 0 0 Rn K R n V2 P n Ω R 0 R n Ω α metal K 28 gyurcsek.istvan@mik.pte.hu

29 Temp Dependency EMF. Windings resistivity of an electric motor is calculated with specific resistivity ρ Cu m instead of the catalog parameter of σ Cu 56 m/ mm 2 given at 25 C ambient temperature. Find the temperature coefficient of chopper (α Cu ) if the operating temperature is 42 C. θ 25 ρ Cu σ Cu Ωmm2 m Ωm ρ(θ) ρ 0 + α θ θ 0 α ρ θ ρ 0 ρ 0 θ θ K α metal K 29 gyurcsek.istvan@mik.pte.hu

30 Temp Dependency EMF.2 Windings conductivity of an electric motor is calculated with specific σ Cu 48 m/ mm 2 instead of the catalog parameter of σ Cu 56 m/ mm 2 given at 25 C ambient temperature. Find the operating temperature. θ ρ ρ 0 α ρ 0 α K ρ(θ) ρ 0 + α θ θ θ 0 + θ gyurcsek.istvan@mik.pte.hu

31 Temp Dependency EMF.3 Calculate the power of the load when v 230 V ambient temperature is 40 C higher than the original θ 0. Resistors are given as the followings. R θ 0 4 Ω, α K, P W, α K Solution v v + v 2 i R + P 2 i i2 R i v + P i i i,2 230 ± A ቊ.33 A R 2() P 2 2 i Ω (short circuit) R 2(2) P 2 2 i Ω (normal load) gyurcsek.istvan@mik.pte.hu

32 Temp Dependency R R + α θ Ω R 2() R 2() + α 2 θ Ω R 2(2) R 2(2) + α 2 θ Ω i i 2 v R 5.9 A P + R 2() 2() v R.5 A P + R 2(2) 2(2) i 2 R 2() i 2 2 R 2(2) W W In case of short circuit the decrease of power is less because R b is dominant with its less temperature dependency. 32 gyurcsek.istvan@mik.pte.hu

33 Temp Dependency EMF.4 Calculate the equivalent temperature coefficient of series resistors if R has α and R 2 has α 2 temperature coefficient. Solution R θ R 0 + α θ R 2 θ R 20 + α 2 θ R 2 θ R θ + R 2 θ R 0 + α θ + R 20 + α 2 θ R 0 + R 0 α θ + R 20 + R 20 α 2 θ R 0 + R 20 + R 0 α θ + R 20 α 2 θ R 0 + R 20 + α R 0 + α 2 R 20 R 0 + R 20 θ R 20 R 0 + R 20 α 2 α R 0 + α 2 R 20 R 0 + R 20 R 2 θ R 20 + α 2 θ 33 gyurcsek.istvan@mik.pte.hu

6 Find the magnetic excitation, the magnetic induction and the magnetic flux in the toroid when N250, d20 mm, D k 80 mm, I720 ma, μ

34 Stationary MF EMF.5 Find the magnetic excitation, the magnetic induction and the magnetic flux in the core. Θ N I A N I 000 H l N I H l A m B μ 0 μ r H 4π T Φ B A μwb EMF.6 Find the magnetic excitation, the magnetic induction and the magnetic flux in the toroid when N250, d20 mm, D k 80 mm, I720 ma, μ r 300 Θ N I A H l N I H B μ 0 μ r H 4π T Φ B A B d2 π μwb 34 gyurcsek.istvan@mik.pte.hu N I l N I D k π 76.2 A m

35 Questions

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