Probabilistic Methods in Engineering
Exercise Set 3
Date Due: 4:00 PM, Wednesday, the 20th of October 2010 Office hours: Tuesdays and Thursdays, 12:00-14:00 and on the SAKAI system You are required to compose your solutions in neat and legible handwriting
VE230 Electromagnetics I
Fall 2009
VE230 Homework 8 solution reference 8.1 Solution
(a) this is a left-hand circularly polarized plane wave
^ ^ Ei ax Ei 0e j1z a y Ei 0e j 1z e j 2
f 200MHz 2f 400106 (rad/s) 1
2
400106 4 (rad/m) c 3 108 3
2
E 5 V/m Ei
VE 230 Electromagnetics I VE230 Homework 7 (Due 12/23/10)
Fall 2010
7.1 Derive the general wave equations for E and H in a nonconducting simple medium where a charge distribution and a current distribution J exist. Convert the wave equations to Helmho
VE 230 Electromagnetics I VE230 Homework 6 (Due 12/07/10)
Fall 2010
6.1 In terms of the d-c current I, how much magnetic energy is stored in the insulating medium of a 3-m-long, air-filled section of a coaxial transmission line, given that the radius
VE 230 Electromagnetics I VE230 Homework 5 (Due 11/26/10)
Fall 2010
5.1 In a cylindrical coordinate system, a 2-m-long straight wire carrying a current of 5 A in the positive z-direction is located at r=4 cm, =pi/2, and -1m z 1m .
^ (a) If B = r 0.2c
VE 230 Electromagnetics I VE230 Homework 4 (Due 11/5/10)
Fall 2010
3.1 The figure below shows three planar dielectric slabs of equal thickness but with different dielectric constants. If Eo in air makes an angle of 45o with respect to the z-axis, find
VE 230 Electromagnetics I VE230 Homework 2 (Due 9/30/10)
Fall 2010
2.1 Coordinate system transformation: (a) transform this vector from Cartesian to cylindrical and evaluate it at the point P1(1, -1, 2).
^ C=x
y2 x2 ^ ^ -y 2 + z 4 x2 + y2 x + y2
2.2
7.1 Solution (1) derive the general wave equations for E and H (nonconducting simple medium)
H t E H = J + t E = -
We take curl of E = -
g E =
Maxwell's equation in simple medium
g H = 0 H E and use H = J + : t t
E = -
E J 2 E H ) = - J + = - - 2 ( t
VE230 Homework6 solution reference 6.1 Solution
0 I 2 r r2 I Ih r 2 0 = hdr = 0 ln r1 2 r 2 r1 2 1 1 0 I h r 2 4 10-7 I 2 3 10 Wm = I = ln = ln = 2.08 10 -7 I 2 ( J ) 2 2 2 r1 4 5 B= ur r B dl = 0 I
6.2 Solution
r u r 2r r ur r 2u V21 = B dl = rw B dl
1
Reference Solutions for HW5 5.1 Solution20' (a) Fm = Idl B .
l
l
r
r
Since B |
r
=
2
= 0 , Fm = 0 (8')
(b) Fm = Idl B =
r
r
1
-1
^ ^5 ^ z r 0.2 cos dz = 2cos (4')
When rotating the wire once about the Z-axis, the work that the magnetic force acting on th
3.1 Solution(16')
r Because of the symmetry of the semicircle, the x component of E at the center vanishes, leaving only y component. dE y = l bd sin (6') 4 0b 2
Figure 1
The electric field intensity at the center is r r r bd r l E = - a y dE y = - a y l
VE 230 Electromagnetics I VE230 Homework 1 (Due 9/17/10)
Fall 2010
^ ^ ^ ^ ^ ^ ^ 1.1 Given vectors A = x 2 + y 4 - z 3 , B = x 2 + y 4 , and C = y 2 - z 4 , find the following: ^ (a) A and a (b) The component of B along C (c) AC (d) A B (e) A (B C ) ^ (g)
2.1 Solution(a): x = r cos - sin , y = r sin + cos , z = z ,
x = r cos , y = r sin , z = z
Substitute them gives:
C = r cos - sin
(
)
r 2 sin 2 r 2 cos 2 - r sin + cos + 4z r2 r2
(
)
= r cos sin ( sin - cos ) - ( sin 3 + cos3 ) + 4 z At point P1, x=1
VE 230 Electromagnetics I VE230 Homework 3 (Due 10/26/10)
Fall 2010
3.1 A line charge of uniform density l in free space forms a semicircle of radius b. Determine the magnitude and direction of the electric field
VE 230 Electromagnetics I VE230 Homework 1 (Due 9/17/10)
Fall 2010
1.1 Given vectors A = x 2 + y 4 - z 3 , B = x 2 + y 4 , and C = y 2 - z 4 , find the following:
(a) A and (b) The component of B along C (c) AC (d) A B (e) A ( B C ) (f) A ( B C ) (g) x B