TUTORIAL 7
(SOLUTION)
EEE 3133 ELECTROMAGNETIC FIELDS AND WAVES DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING
1.
The conducting triangular loop in Figure 1 carries a current of 10 A. Find H at (0,0,5) due to: a. Side 1 of the loop b. Side 3 of the loop
Figure 1 Ans: H1 = -59.1 ay mA/m, H2 = -30.63a x + 30.63 ay mA/m Solution:
a. IJ# IJ% IJ $ $$$ 0 #" { $ ){ ) -%.1 H & {IJ$ IJ#)&{') $$ mA/m b. IJ# IJ% IJ $ $$ 0
#" { $$ ){ $ ) H & {IJ$ IJ#)&{') -3.3 3. mA/m 2.
A thin ring of radius 5cm is placed on plane z = 1cm so that its center is at (0,0,1 cm). If the ring carries 50 mA along a φ, find H at: a. (0,0,-1cm) b. (0,0,10 cm) Ans: (a)400 az mA/m, (b)57.3 az mA/m Solution:
a.
H ${)
{ {1)1) I ){01&) {01 H {$ $)$ 0 1 / / . . m/ b.
{1) %I ){01&) {01 H {$ %$)$ 0 1 / / .33 m/
3.
Plane y=1 carries current K = 50 az mA/m. Find H at: a. (0,0,0) b. (1,5,-3) Ans: (a)25 ax mA/m, (b)-25 ax mA/m Solution:
a. b.
4.
H #$ 0 H{)) #$ { 0{) m H{1 H{1 3) 3) #$ { 0 {) m
A/m A/m
An infinitely long transmission line whose cross section in Figure 2 consisting of two concentric cylinders having their axes along the z-axis. The inner conductor has radius, a and carries current I while the outer conductor has inner radius, b and thickness, t carries return current, -I . Assuming the current is uniformly distributed in both conductors, determine the expression of H by applying Ampere’s Law along the Amperian path of the following regions: i.
0 ≤ ρ ≤ a
ii.
a ≤ ρ ≤ b
iii.
b ≤ p ≤ b+t
iv.
ρ ≥
b+t
Figure 2
Solution: a. Region 0 ≤ ρ ρ ρ ρ ≤ a
H Ñ Ñ
$ $ $ $ Ñ $ $ $ (" (" $ H H $ $ H $ b. Region a ≤ ρ ρ ρ ρ ≤ b
H Ñ H H c. Region b ≤ p ≤ b+t
H Ñ H Ñ
Ñ { { )$ $ $
$ $ { 1 $ { )$ $ (" ( $ $ H 1 $ d. Region ρ ρ ρ ρ ≥ b+t
H Ñ {) {)
5.
2
2
A current distribution gives rise to the vector magnetic potential A = x y ax + y x ay – 4xyz az Wb/m. Calculate the following: a. B at (-1,2,5) b. The flux through the surface defined by z = 1, 0 ≤ x≤ 1, -1 ≤ y ≤ 4 2
Ans: (a)20 ax + 40 ay + 3 az Wb/m , (b)20 Wb Solution:
a. 0 { { ) {){$ $) {1) 3Wb/m & # { $ $) b. Ñ (# (" & # ${) $ (# (" 13 {1){ 3) Wb ψ
