CHAPTER 5
5.1. Given the current density J = 104 [sin(2x)e2y ax + cos(2x)e2y ay ] kA/m2 :
a) Find the total current crossing the plane y = 1 in the ay direction in the region 0 < x < 1,
0 < z < 2: This is found through
ZZ
Z 2Z 1
Z 2Z 1
I=
J n da =
J ay
d
CHAPTER 1
1.1. Given the vectors M = 10ax + 4ay 8az and N = 8ax + 7ay 2az , nd:
a) a unit vector in the direction of M + 2N.
M + 2N = 10ax 4ay + 8az + 16ax + 14ay 4az = (26, 10, 4)
Thus
a=
(26, 10, 4)
= (0.92, 0.36, 0.14)
(26, 10, 4)
b) the magnitude of
CHAPTER 2 2.1. Four 10nC positive charges are located in the z = 0 plane at the corners of a square 8cm on a side. A fifth 10nC positive charge is located at a point 8cm distant from the other charges. Calculate the magnitude of the total force on this fi
CHAPTER 1 1.1. Given the vectors M = 10ax + 4ay  8az and N = 8ax + 7ay  2az , find: a) a unit vector in the direction of M + 2N. M + 2N = 10ax  4ay + 8az + 16ax + 14ay  4az = (26, 10, 4) Thus a= b) the magnitude of 5ax + N  3M: (5, 0, 0) + (8, 7,
CHAPTER 1
1.1. Given the vectors M = 10ax + 4ay 8az and N = 8ax + 7ay 2az , nd:
a) a unit vector in the direction of M + 2N.
M + 2N = 10ax 4ay + 8az + 16ax + 14ay 4az = (26, 10, 4)
Thus
a=
(26, 10, 4)
= (0.92, 0.36, 0.14)
(26, 10, 4)
b) the magnitude of
CHAPTER 2
2.1. Three puiut charges are pusitieneti in the .ry plane as I'elltm's'. 511C at. y = 5 1111, 111 [1'3 :11 y = 5 em.
15 11? at .r = .'1 cm. Finr1 the required .ry emrriinates sf :1 211110 fetlrth charge that. Twill protime 11
zero electric el
CHAPTER 3
3.1. Suppose that the Faraday concentric sphere experiment is performed in free space using a
central charge at the origin, Q1 , and with hemispheres of radius a. A second charge Q2 (this
time a point charge) is located at distance R from Q1 , w
CHAPTER 4
4.1. The value of E at P ( = 2, = 40 , z = 3) is given as E = 100a 200a + 300az V/m.
Determine the incremental work required to move a 20 C charge a distance of 6 m:
a) in the direction of a : The incremental work is given by dW = q E dL, where
CHAPTER 6.
6.1. Consider a coaxial capacitor having inner radius a, outer radius b, unit length, and lled with
a material with dielectric constant, r . Compare this to a parallelplate capacitor having plate
width, w, plate separation d, lled with the sam
CHAPTER 7
7.1a. Find H in cartesian components at P (2, 3, 4) if there is a current lament on the z axis carrying
8 mA in the az direction:
Applying the BiotSavart Law, we obtain
Z
Z
Z
I dL aR
I dz az [2ax + 3ay + (4 z )az ]
I dz [2ay 3ax ]
Ha =
=
=
4 R2
DIFFERENTIAL EQUATIONS
DEPARTMENT OF PHYSICS
WESTMONT COLLEGE
PROBLEM SET NO. 11
CHAPTER 7:
THE LAPLACETRANSFORM METHOD
SECTION 7.1:
Definition of the Laplace Transform.
EXERCISES:
1, 3, 6, 18, 25, 38, 41, 42.
SECTION 7.2:
Inverse Transforms and Transfor
DIFFERENTIAL EQUATIONS PHY040
DEPARTMENT OF PHYSICS
WESTMONT COLLEGE
DIFFERENTIAL EQUATIONS PHY040 MIDTERM EXAM I
CHAPTER 1: INTRODUCTION TO DIFFERENTIAL EQUATIONS
1.1 Definitions and Terminology
Identification of the type, order, and linearity (homogen
ECE 257 : Engineering Electromagnetics
Design Problems
Date of allotment:14/3/09
Date of Submission:20/3/09
1. Imagine that new and extraordinarily precise measurement have revealed an error
in Coulombs law. The actual force of interaction between two poi
5Assignment 1
ECE 257 Engineering Electromagnetics
DOA: 23/2/09
DOS:2/03/09
Q1. Two vectors are represented by: A = 2ax + 2ay + 0az and B = 3ax + 4ay + 2az .Find the
dot and crossproduct of the angle between the vectors.Show that AXB is at right angle to
CHAPTER 3 3.1. An empty metal paint can is placed on a marble table, the lid is removed, and both parts are discharged (honorably) by touching them to ground. An insulating nylon thread is glued to the center of the lid, and a penny, a nickel, and a dime
CHAPTER 4 4.1. The value of E at P ( = 2, = 40 , z = 3) is given as E = 100a  200a + 300az V/m. Determine the incremental work required to move a 20 C charge a distance of 6 m: a) in the direction of a : The incremental work is given by dW = q E dL, whe
CHAPTER 5 5.1. Given the current density J = 104 [sin(2x)e2y ax + cos(2x)e2y ay ] kA/m2 : a) Find the total current crossing the plane y = 1 in the ay direction in the region 0 < x < 1, 0 < z < 2: This is found through I= J n da =
S 0 1 2 0 1
S
J ay
y=
CHAPTER 6. 6.1 Construct a curvilinear square map for a coaxial capacitor of 3cm inner radius and 8cm outer radius. These dimensions are suitable for the drawing. a) Use your sketch to calculate the capacitance per meter length, assuming R = 1: The sket
CHAPTER 7 7.1. Let V = 2xy 2 z3 and = 0 . Given point P (1, 2, 1), find: a) V at P : Substituting the coordinates into V , find VP = 8 V. b) E at P : We use E = V = 2y 2 z3 ax  4xyz3 ay  6xy 2 z2 az , which, when evaluated at P , becomes EP = 8ax +
CHAPTER 8 8.1a. Find H in cartesian components at P (2, 3, 4) if there is a current filament on the z axis carrying 8 mA in the az direction: Applying the BiotSavart Law, we obtain Ha =

I dL aR = 4R 2

I dz az [2ax + 3ay + (4  z)az ] = 4(z2  8z +
CHAPTER 9 9.1. A point charge, Q = 0.3 C and m = 3 1016 kg, is moving through the field E = 30 az V/m. Use Eq. (1) and Newton's laws to develop the appropriate differential equations and solve them, subject to the initial conditions at t = 0: v = 3 105
CHAPTER 10 10.1. In Fig. 10.4, let B = 0.2 cos 120t T, and assume that the conductor joining the two ends of the resistor is perfect. It may be assumed that the magnetic field produced by I (t) is negligible. Find: a) Vab (t): Since B is constant over the
CHAPTER 11
11.1. Show that Exs = Aej k0 z+ is a solution to the vector Helmholtz equation, Sec. 11.1, Eq. (16), for k0 = 0 0 and any and A: We take d2 2 Aej k0 z+ = (j k0 )2 Aej k0 z+ = k0 Exs dz2 11.2. Let E(z, t) = 200 sin 0.2z cos 108 tax + 500 cos(0.
CHAPTER 12
+ + 12.1. A uniform plane wave in air, Ex1 = Ex10 cos(1010 t z) V/m, is normallyincident on a copper surface at z = 0. What percentage of the incident power density is transmitted into the copper? We need to find the reflection coefficient. T
CHAPTER 13 13.1. The parameters of a certain transmission line operating at 6 108 rad/s are L = 0.4 H/m, C = 40 pF/m, G = 80 mS/m, and R = 20 /m. a) Find , , , , and Z0 : We use = ZY = (R + j L)(G + j C) = [20 + j (6 108 )(0.4 106 )][80 103 + j (6 108 )
CHAPTER 14 14.1. A parallelplate waveguide is known to have a cutoff wavelength for the m = 1 TE and TM modes of c1 = 0.4 cm. The guide is operated at wavelength = 1 mm. How many modes propagate? The cutoff wavelength for mode m is cm = 2nd/m, where n is
CHAPTER 1 1.1. Given the vectors M = 10ax + 4ay  8az and N = 8ax + 7ay  2az , find: a) a unit vector in the direction of M + 2N. M + 2N = 10ax  4ay + 8az + 16ax + 14ay  4az = (26, 10, 4) Thus a= b) the magnitude of 5ax + N  3M: (5, 0, 0) + (8, 7,
DIFFERENTIAL EQUATIONS PHY040
DEPARTMENT OF PHYSICS
WESTMONT COLLEGE
DIFFERENTIAL EQUATIONS PHY040 MIDTERM EXAM II
CHAPTER 4: HIGHERORDER DIFFERENTIAL EQUATIONS
4.1 Preliminary Theory
4.1.1 Preliminary Theory
What is an InitialValue Problem? What is a