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UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING

ECE 1 IOH1 S -- ELECTRICAL FUNDAMENTALS FINAL EXAMINATION, APRIL 24,2017,2:00 pm

First Year -- Computer, Electrical, Industrial, Mechanical, Materials, and Track One Engineering Programs.

Examiners S. Aitchison, B. Bardakjian, A. Flelmy, M. Mojahedi, K. Truong and P. Yoo

(e= I.6x 10-19 C, Co=8.85 x 10-12 F/rn, to=4itx 10 H/rn, g=9.81 N/kg)

NAME: (PLEASE PRINT) Family (Last) Name Given (First) Name

STUDENT NUMBER:

EXAMINATION TYPE CALCULATORS

DURATION: INSTRUCTIONS:

A Casio FX-991MS & Sharp EL-520X 2.5 hours

Answer all five questions. Put the answers in the boxes provided. For multiple choice questions please circle 1 of the answers All work is to be done on these pages. Show steps, compute numerical results when requested and state units. Write down any assumption made. You may use the back of each page. Last blank page may be removed for rough work.

Question Mark

1

2

3

4

5

Total

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QI 110 marks] Electrostatics

(a) 12 marks] The figure shows the cross section of a metallic shell with inner radius r ju and outer radius rou t. The shell is made of perfect metal and is charged with the total charge of -8 nC. A positive charge of +3 nC is placed at the center of the shell. Circle the correct statement regarding the charges appearing at r = r 71 , r1 < r <r0, and r =

Charges at r = r 1 , r1 <r <r0 , and r = rout are respectively, +3 nC, 0, and -5 ,i(7.

Charges at r = r 1 , r1 <r < rout, and r = rout are respectively, -3 n(I, 0, and -5 nC.

Charges at r = r 171 , r 1 <r < rout, and r = rout are respectively, -3 n(.7, +3, and -5 nC.

Charges at r = r1, rin < r < rout, and r = rout are respectively, -3 nC, 0 and -8 nC,

(b) 12 marks] The figure shows a static electric field, E. An electron (e) moves from the position marked (i) to the position marked (. Circle the correct statement regarding the potential energy (U) and electrostatic potential (I") of the electron.

e

Potential energy (U) has increased, but electrostatic potential (V) has decreased

Both potential energy (U) and electrostatic potential (V) have decreased

Potential energy (U) has decreased, but electrostatic potential (I') has increased

Both potential energy (U) and electrostatic potential (I") have increased

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(c) 12 marks] A dielectric with k = 4, is inserted into a parallel plate capacitor such that it fills 1/3 of the space as shown in the figure below. If the capacitance of the parallel plate capacitor

before the dielectric was inserted was C, what is the new value of capacitance.

4

0.5 C

0.75 C

C

2 C

6 C

(d) [2 marks] Consider the two infinite sheets of charge, separated by a distance d shown below.

Each sheet has a charge density of + u. A point charge of + 2Q is placed at a distance d, above the top sheet. Which expression describes the electric field at point A, midway between the two sheets.

+2Q

+0

imi

i)E=

E = 0

E = 1 8Q

41ts0 9d 2

E= E() 9d 2

v)E= E0 9d2

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Qi continued

(e) 12 marks] Consider a capacitor of length L that is composed of two concentric conducting cylinders with radii a and b as shown in figure below. Which of the following is the correct expression for the capacitance of these two cylinders?

C = 27r 0Ln()

C = VQ

iil)C= 27rEO L—)

iv) C= Ln()_!_L

v)C= 2ir0L—

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Q2 110 marks] Magnetic Fields

(a) 12 marks] Consider two wires, carrying currents i and 4i out of the page as shown in the figure below.

H 4i

d

The wires are separated by a distance of d = 5 cm. At what position is the magnetic field zero.

i) (-1 ciii, 0)

H) (I cm, 0)

(4 cm. 0)

(2 cm. 0)

(2.5 cm, 0)

(b) 12 marksj For the two long straight wires carrying currents i1 and i2 respectively as shown below, which one of the following represents the expression for the magnetic field at the origin.

V

X

i11012 I 2ird 2

ii)1f 2ird1

9012 Poll 2ivd2 2ird

iv)i + .!Q!L 21rd2 27td1

V) /1012 - I+ 11011 -j 2gd2 2ird1

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Q2 continued,

(c) 12 marksl The schematic in the figure below shows the cross-section of a metallic wire with a radius R, which carries a uniform current I, out of the page. Which of the following is an expression for the Magnetic Field, B at a radius, r < R?

i)B= poi

2Tr

B = r Poi (__) \2irR2

B = Zivr 2

B = R

2irtR 2 v)B=

ON

r

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Q2 continued.

(d) 12 marks] The uniform magnetic field B as shown below, changes over time according to the

following equation: B (t) = 20 On

sin (27rt) where B is in Tesla and tis in second. Which of the

following is the magnetic flux () as a function of time, which passes through a 1 cm x 1 cm square-shaped wire loop.

8 wire loop

t = f / 10 sin (27rt) 7

B = uj\

cos (2mt) \2001r)

io cos(27rt)

iv) 4DB = — (—i--" sin (27rt) 2OOir)

V) (PB = (701-0;) 10 5cos (2irt)

(e) 12 marks] Consider the wire, which carries a current of 2i as shown in the figure below. What is tie magnitude of the magnetic field at point P?

2i 0.

i)B=

fl) B = "01 Br

B = 2irr

B = LIL 4rr

V) B = C'

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Q3 [10 marks] Consider the following circuit below,

:TT

5mA

1 k

(a) 14 marks] Find the Thévenin equivalent of circuit A

Thévenin Equivalent Circuit:

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Q3 continued. (Circuit diagram has been duplicated for your convenience)

2k0

2k.0 ± 1O\J 2mA 4k0

2k0: 2k0 <6V

1ECIrcuit

- v,, + RL rCircuit Al iB

F kO 1kO 3V

(b) [4 marks] Find the Thévenin equivalent of circuit B

Thévenin Equivalent Circuit:

(c) [2 marks] Determine RL that yields maximum power transfer from all sources in circuits A & B

RL=

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Q4 [10 marksi Consider the following circuit below, the switch is in position A for a long time. At t = 0, the switch is moved to position B.

in J4B

VW

1 A 4 02 F T

- t)

10 V 3

6 fl 80 (a) 16 marks] Derive the function v(t) for t ? 0.

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Q4 continued. (Circuit diagram has been duplicated for your convenience)

10 50 30

VW

1 A7470-4 TV

C(t)

+

iov 30

60 80

(b) 14 marks] Derive the function v(t) for t? 0.

5H by

Ii

v (t

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QS 110 marks] Consider the following circuit below,

Where v(t) = 10 sin (200t) in V, and i(t) = 5 cos(200t) in mA.

(a) 12 marks] Compute the contribution of the DC voltage source to the output voltage v0 (t). HINT: Use source superposition to compute v0 (t)

DC V (t) =

(b) 12 marks] Compute the impedances.

Zcap =

Zinci =

5H 1OV

t RM

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Q5 continued. (Circuit diagram has been duplicated for your convenience)

(c) 1.6 marks] Compute the contribution of the AC sources to the output voltage v0 (t).

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