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Unformatted text preview: Mark
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/10 UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION, APRIL 2006
First Year  Engineering Science ECE 159$  ELECTRICITY & ELECTRIC CIRCUITS Exam Type: A
TOTAL /100 Examiners: N.P. Kherani and B. Wang NAME: Last First STUDENT NO: INSTRUCTIONS: o This is a Type A examination; no aids are allowed.  Only nonprogrammable calculators are allowed.  Answer all parts of all ten questions.  All ten questions are of equal weight, though not of equal level of difficulty.
o All work is to be done on the pages of this booklet. o When answering the questions include all the steps of your work on these
pages. For additional space, you may use the back of the preceding page. . Do not unstaple this exam booklet. CONSTANTS: , e = 1.6x10'19 0 _ .9¢,=8.85x10‘12 02/(Nm2) ,tlo=47rx10'7 Tm/A Page 1 of 12 . Name: , Student No.: QUESTION 1 [10 marks] Fill in the blanks, and circle TRUE or FALSE, as appropriate. Comment if you must. Each part
is worth 1 mark. v a) Name one example of natural electromagnetic phenomena: b) Charles Augustin de Coulomb invented the torsion balance in 1784. He used this for the measurement of electrical forces. Some years later, Cavendish used a torsional balance to measure the gravitational constant G. TRUE / FALSE c) Two charges of equal magnitude and opposite sign separated by the distance 2d is an example of an ’ d) Give two examples of dielectrics: (i)  (ii) e) What is the dielectric strength of vacuum? , (include the
units) f) The magnetic ﬁeld does perform work. TRUE / FALSE g) Law is vital for the production of electrical power. h) Kirchhost Current Law is another statement of i) The voltage across a capacitor can change instantaneously. TRUE / FALSE V j) At resonance the total reactance in the circuit is Page 2 of 12 . Name: Student No.2 QUESTION 2 [10 marks] Consider the parallel plate capacitor shown below. The conducting plates are of area A each and
have charge Q and —Q, respectively. The plates are separated by a distance d. The space between the plates is ﬁlled with a dielectric material such that its dielectric constant, K=K(x), has a linear
dependence on x as shown below. a) Find the surface charge density la] . [1 mark] b) Find the linear expression for K=K(x). [1 mark] c) Using Gauss’ law, @306 d;1 = q , ﬁnd the electric ﬁeld strength E(x) forx ranging from 0
to d. [3 marks] d) Find the potential difference V(x=0)  V(x=d). [3 marks] c) Find the capacitance of the parallel plate capacitor. [2 marks] Page 3 of 12 Name: Student No.2 QUESTION 3 [10 marks] Consider a coaxial cable which is made of two concentric, long, cylindrical “pipes”. The inner
pipe has a radius R, while the outer pipe is of radius R2. Both pipes are of negligible thickness. The inner and outer pipes each carry a current 1(t) = losin(a)t) in opposite directions. A section of the cable of length L with the directions of current is shown below. M) (on the outer pipe) [(0 (on the inner pipe) a) Find the magnitude of the magnetic ﬁeld B(r,t) in the annular region R, < r < R 2. Also, state
the direction of the magnetic ﬁeld (clockwise or counter clockwise). [6 marks] b) Using the result from part (a) and with reference to the ﬁgure on the right, ﬁnd the
induced emf arOund the path ABCD as a
function of t. [4 marks] Page 4 of 12 . Name: Student No.: QUESTION 4 [10 marks] Consider two identical parallel plate capacitors. The plate area is A and plate separation is d. The ﬁrst capacitor has charges Q and Q on its two plates, respectively. The second capacitor has no ' charge on its plates. The capacitors are isolated from each other, as shown below. Q_Z_ _L id id Q
Capacitor 1 Capacitor 2
a) Calculate the capacitance of both capacitors, C I and C2. [2 marks]
b) Find the potential difference across both capacitors, V1 and V2. [2 marks]
c) Calculate the electrical energy stored in both' capacitors, W1 and W2, and hence the total energy WT= W, + W2 . [2 marks] (1) Now, the two capacitors are connected with copper wires as shown below. What is the
charge on the plates of the ﬁrst and second capacitors? Also, calculate the total electrical
energy stored on the two capacitors. Is this result different from that obtained in part (c), and if so why? [4 marks] wire A
V Capacitor 1 Wire Capacitor 2 Page 5 of 12 Name: Student No.2
QUESTION 5 [10 marks] Using Nodal Analysis ﬁnd the current labelled "I" in the circuit shown below. Page 6 of 12 ———__ Name: Student No.2 QUESTION 6 [10 marks] Consider the following circuit which contains terminals A and B. 8D 24D
12V a) Find the Thévenin equivalent resistance for the above circuit with respect to terminals A and
B. [3 marks] ‘ b) Find the Thévenin equivalent source with respect to terminals A] and B. [4 marks] c) Draw the Thévenin equivalent circuit with respect to terminals A and B. [1 mark] (1) Calculate the power delivered to a 90 load when connected to terminals A and B. [2 marks] Page 7 of 12 —————_—_ Name: Student No.: QUESTION 7 [10 marks] Consider the following circuit which contains an ideal operational ampliﬁer. Find the power delivered to the load resistor, RL. Page 8 of 12 Name: Student No.1 QUESTION 8 [10 marks] The switch in the RL circuit shown below has been open for a long time. At t = 0 the switch is closed. 29 4Q a) b) C) Find the current just before the switch closes, i,,(0'). [1 mark] Find the current just after the switch closes, io(0+). [4 marks] Find the current io(t) for t > O. [5 marks] Page 9 of 12 Name: Student No.: QUESTION 9 [10 marks]
Question 9 a) Consider the AC circuit shown on
the right. Write the nodal equations for the node voltages V1 and V2 .
[2 marks] b) Solve for V2 in terms of VS andﬁ using the equation at V2 . [2 marks] c) Solve for V1 in terms of Xi and V2 using the equation atﬁ. [2 marks] Page 10 of 12 , Name: Student No.: QUESTION 9 (. .. continued) (1) Solve forﬁ in terms of VS . [2 marks] ‘ V
6) Find the input impedance Z in = 7—S— . [2 marks]
S Page 11 of12 Name: Student No.2 ————_—_ QUESTION 10 [10 marks] An electric motor rated at 1500 W is powered by a 240 Vrms, 60 Hz source. The motor is
operating at a lagging power factor of 0.85. a) Assume that the motor can be represented by a resistor connected in series with an '
inductor, calculate the corresponding resistance and inductance. [5 marks] b) What component could you connect in parallel with the motor in order to Operate the
motor at a power factor of unity? What is the value of this component? [5 marks] Page 12 of12 ...
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