Download final examination - McMaster University > ECE

Student Name:
Student ID:
Instructor: Natalia K. Nikolova
COURSE ELECTRICAL ENGINEERING 2FH3
Duration of Examination: 2.5 hours
McMaster University Final Examination
April 20, 2016
THIS EXAMINATION PAPER INCLUDES 6 PAGES, 7 MANDATORY QUESTIONS and 1 BONUS
QUESTION. YOU ARE RESPONSIBLE FOR ENSURING THAT YOUR COPY OF THE PAPER IS
COMPLETE. BRING ANY DISCREPANCY TO THE ATTENTION OF YOUR INVIGILATOR.
Instructions:
1. You can use only a standard calculator (Casio-FX991).
2. Write your name and student ID on each page, the exam booklets incl.
3. You are allowed to bring 1 sheet of letter-size paper with any writing
on both sides of the sheet.
4. Answer questions 1 TO 7. Provide detailed solutions in the exam booklet.
5. Question 8 is a bonus question. Answering it is optional.
Question 1 [34 points] (Homework 02, 2.13)
A uniform volume charge density of ρν = 5 μC/m3 is present throughout the spherical shell
extending from r1 = 3 cm to r2 = 5 cm. Elsewhere, ρν = 0. Medium is vacuum.
(a) Find the total charge Q in the shell. [4 points]
(b) Find the E-field vector at the point A with coordinates rA = 1 cm, θA = 90°, φA = 0°. Give the
answer using spherical components. [2 points]
(c) Find the E-field vector at the point B with coordinates rB = 4 cm, θB = 90°, φB = 90°. Give the
answer using spherical components. [4 points]
(d) Find the E-field vector at the point C with coordinates rC = 6 cm, θC = 0°, φC = 0°. Give the
answer using spherical components. [3 points]
(e) Find the absolute potential VC at point C, provided the potential is assumed zero at infinity. [3
points]
(f) Find the voltage VCD between the points C and D where rD = 10 cm, θD = 0°, φD = 0°. [3
points]
(g) Write a matlab code that computes VC at point C using the principle of superposition. Do not
worry about the syntax; syntax errors will be ignored. [15 points]
Question 2 [14 points] (Tutorial 11, Q2)
A parallel-plate capacitor is made using two circular plates of radius a = 2 mm, with the bottom
plate on the xy plane (z = 0), centered at the origin. The top plate is located at z = 0.5 mm, with its
center on the z axis. Potential V0 = 5 V is applied to the top plate whereas the bottom plate at z = 0
is grounded (its potential is zero). Dielectric having radially dependent permittivity fills the region
between plates. The relative permittivity is given by εr(ρ) = 1 + ρ/a. Find:
(a) the absolute value of the charge Q on each electrode; [10 points]
(b) the capacitance C; [2 points]
(c) the electric energy We stored in the capacitor. [2 points]
Question 3 [10 points] (Homework 13, 5.34 8th ed)
Region 1 (x ≥ 0) is a dielectric with relative permittivity εr1 = 1, while in region 2 (x < 0) εr2 = 5.
Let E1 = 20ax − 10ay + 50az V/m.
(a) Find the electric field vector E2 in region 2. [4 points]
(b) Find the polarization vector P2 in region 2. [3 points]
(c) Find the absolute value of the surface bound charge density |ρsb| at the interface x = 0. [3 p.]
Page 1 of 6, continued
Student Name:
Student ID:
Question 4 [14 points] (similar to L10, Example sl. 9)
A resistor is made of two coaxial cylindrical electrodes of radii a =
0.5 mm and b = 1 mm. Its cross-section is shown in the figure. The
resistor has length l = 2 mm. The medium between the electrodes has
specific conductivity σ = 50 S/m. The total current flowing through
the resistor is I = 2 A.
(a) Find the current density J as a function of the distance ρ to the
axis of the inner cylinder. [5 points]
(b) Find the electric field vector E as a function of the distance ρ to
the axis of the inner cylinder. [2 points]
(c) Find the voltage between the two electrodes. [3 points]
(d) Find the resistance of this cylindrical resistor. [2 points]
(e) Find the power dissipated in the resistor. [2 points]
b
σ
a
I
Question 5 [6 points] (L16, sl. 17)
Surface current sheet of current density K = 4ax A/m lies in the plane defined by y = 4 mm. Find
the magnetic field H at the origin (0,0,0).
Question 6 [10 points] (L17, sl. 10)
The magnetic field is given as
5
H( ρ ,φ ) = sin(0.1φ )a ρ A/m.
ρ
Find the current density J at the point P with coordinates ρP = 1 mm, φP = 0°.
Question 7 [12 points] (Homework 15, 7.9)
(a) [10 points] Find the inductance of a toroid consisting of 200 turns. Its cross-section is of
width w = 1 cm and height h = 1 cm. Its inner radius is ρ1 = 1.5 cm. The toroid has a ferromagnetic
core whose relative permeability is μr = 400. Take into account the internal wire inductance in
addition to the inductance due to the field inside the core.
(b) [2 points] What is the energy stored in this inductor if the current through it is I = 1 A?
Question 8 [2 bonus points]
Electromagnetic energy storage always employs capacitive structures (e.g., batteries) to store
energy for long periods of time. Why not inductors? If you provide several reasons, please list them
in the order of their significance.
END OF QUESTION SHEET
A sheet of mathematical formulas follows (4 more pages)
TOTAL MARKS FOR THIS EXAM = 100
Page 2 of 6, continued