DR. GYURCSEK ISTVÁN Exercises in Electromagnetic Field Sources and additional materials (recommended) Gyurcsek I. Elmer Gy.: Theories in Electric Circuits, Globe Edit 206, ISBN:97833307343 Simonyi K.: Villamosságtan. AK Budapest 983, ISBN:963053434 gyurcsek.istvan@mik.pte.hu 208.07.09.
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 4 9.625 mm 2 R ρ l A 0.025 0. 9.625.2739 0 4 27.39 μω I V R 0 3.2739 0 4 7.85 A 2 gyurcsek.istvan@mik.pte.hu 208.07.09.
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 208.07.09.
Stationary Electric Field R ABm ρ l A R CDm ρ l A 4 gyurcsek.istvan@mik.pte.hu 208.07.09.
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.74 0 7 Ω I ABm V R ABm 5.834 ka 5 gyurcsek.istvan@mik.pte.hu 208.07.09.
Stationary Electric Field l m d m π 2 d k + d b 2 2 π 204.2 mm 0.2042 m A CD h d k d b 2 3500 mm 2 R CDm ρ l m A CD.46 0 6 Ω I CDm U R CDm 685 A 6 gyurcsek.istvan@mik.pte.hu 208.07.09.
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 208.07.09.
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 2.5 0 8 0.05 π ln 200 60.962 0 7 Ω I AB V R AB 5,29 A 5,834 A 8 gyurcsek.istvan@mik.pte.hu 208.07.09.
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 208.07.09.
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.5 2.5 0 8 π ln 200 60 7.665 05 S I CD V R CD R CD G CD.3046 0 6 Ω 0 3.3046 0 6 766 A 685 A 0 gyurcsek.istvan@mik.pte.hu 208.07.09.
Stationary Electric Field EMF.03 Calculate the earth resistance when the ground is wet and when it is dry. gyurcsek.istvan@mik.pte.hu 208.07.09.
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 208.07.09.
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 000 3.83 3.83 kv V fdry I R fdry 000 38.3 38.3 kv 3 gyurcsek.istvan@mik.pte.hu 208.07.09.
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 208.07.09.
Stationary Electric Field R,2WET 2π σ WET r r 2 2π 0..5 0.53 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 0 3.5 53. Ω V,2WET I R,2WET 000 0.53 53 V V,2DRY I R,2DRY 000 53. 53. kv 5 gyurcsek.istvan@mik.pte.hu 208.07.09.
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 208.07.09.
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 383 000 3.83 MW P DRY V fdry I 38300 000 38.3 MW 7 gyurcsek.istvan@mik.pte.hu 208.07.09.
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 000 2 0. 2π 0.5 3.83 MW P DRY r 2 dr I 2 2π σ r G 000 2 0 3 2π 0.5 38.3 MW 8 gyurcsek.istvan@mik.pte.hu 208.07.09.
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 6 0-8 / m 2 3 0-9 / m d mm d 2,5 mm A 8 cm 2 A 2 2 cm 2. 9 gyurcsek.istvan@mik.pte.hu 208.07.09.
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 7.98 nw P a P a + P a2 7.98 + 539.4 557.4 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 a2 46.45 0 9 2 250 0 6 539.4 nw A + ε d A + σ d ε 2 A d 2 σ 2 A d 2 20 gyurcsek.istvan@mik.pte.hu 208.07.09. 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
Stationary vs. Static EF R b R b R b2 I b V R b 2 48.45 d σ A 0.2477 μa d σ 2 A 2 2.5 0 3 6 0 8 8 0 4 2.5 0 3 3 0 9 2 0 4 52.08 0 6 694.4 0 6 48.45 MΩ P b P b + P b2 V2 R b + V2 R b2 22 52.08 + 22 694.4 2.765 + 0.2074 2.9724 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 208.07.09.
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 2 5 0-9 / m V 7 kv 22 gyurcsek.istvan@mik.pte.hu 208.07.09.
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 0 3.5 2 ln ln 0.5 +.5 0 8 5 0 9 26.642 kω 23 gyurcsek.istvan@mik.pte.hu 208.07.09.
Stationary vs. Static EF I V R 7 26.642 262.7 ma, P V I 7000 0.2627 839 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 208.07.09.
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 2 5 0-9 / m V 7 kv 25 gyurcsek.istvan@mik.pte.hu 208.07.09.
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 0.5.5 0 8 +.5 2 5 0 9 3.29 MΩ 26 gyurcsek.istvan@mik.pte.hu 208.07.09.
Stationary vs. Static EF I V R 7 03.329 0 7 5.2647 ma, P V I 7 03 5.2647 0 3 36.853 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 208.07.09.
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 0 2400 20 4.2 0 3 K R n V2 P n 2302 00 529 Ω R 0 R n 529 48. Ω α metal 4 0 3 K 28 gyurcsek.istvan@mik.pte.hu 208.07.09.
Temp Dependency EMF. Windings resistivity of an electric motor is calculated with specific resistivity ρ Cu.904 0-8 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 56 0.07857 Ωmm2 m.7857 0 8 Ωm ρ(θ) ρ 0 + α θ θ 0 α ρ θ ρ 0 ρ 0 θ θ 0.904 0 8.7857 0 8.7857 0 8 42 25 3.9 0 3 K α metal 4 0 3 K 29 gyurcsek.istvan@mik.pte.hu 208.07.09.
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 4 0 3 K ρ(θ) ρ 0 + α θ 48 56 4.67 56 4 0 3 θ θ 0 + θ 25 + 4.67 66.67 30 gyurcsek.istvan@mik.pte.hu 208.07.09.
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 0.002 K, P 2 300 W, α 2 0.004 K Solution v v + v 2 i R + P 2 i i2 R i v + P 2 0 4 i 2 230 i + 300 0 i,2 230 ± 2302 4 4 300 2 4 56.6 A ቊ.33 A R 2() P 2 2 i 300 56.6 2 0.095 Ω (short circuit) R 2(2) P 2 2 i 300 68.24 Ω (normal load) 2.332 3 gyurcsek.istvan@mik.pte.hu 208.07.09.
Temp Dependency R R + α θ 4 + 0.002 40 4.32 Ω R 2() R 2() + α 2 θ 0.095 + 0.004 40 0. Ω R 2(2) R 2(2) + α 2 θ 68.24 + 0.004 40 95.6 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) 297.28 W 259.44 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 208.07.09.
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 208.07.09.
Stationary MF EMF.5 Find the magnetic excitation, the magnetic induction and the magnetic flux in the core. Θ N I 000 000 A N I 000 H l N I H l 4 40 2.5 0 2 666.67 A m B μ 0 μ r H 4π 0 7 000 666.67 0.838 T Φ B A 0.838 0 4 83.8 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 250 0.72 800 A H l N I H B μ 0 μ r H 4π 0 7 300 76.2 0.7 T Φ B A B d2 π 4 36.76 μwb 34 gyurcsek.istvan@mik.pte.hu 208.07.09. N I l N I D k π 76.2 A m
Questions 35 gyurcsek.istvan@mik.pte.hu 208.07.09.