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Unformatted text preview: Page I of 12 UNIVERSITY OF TORONTO
FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION APRIL 23, 2002 FIRST YEAR PROGRAM: ENGINEERING SCIENCE ECE 1598: FUNDAMENTALS OF ELECTRICITY & ELECTRIC CIRCUITS EXAMINERS: M.L.G. Joy and S. Zukotynski (Coordinator) NAME:
L25! First STUDENT NO.: INSTRUCTIONS: 0 This is a Type A examination; no aids are allowed. 0 Only non—programmable calculators are allowed.  Answer all parts of all ﬁve questions. 0 All ﬁve questions are of equal weight. 0 The weight of each of the individual parts of each question is stated in the margins.
0 All work is to be done on these pages. 0 Place your ﬁnal answers in the provided boxes unless instructed otherwise. 0 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. 0 Do not unstaple this exam. CONSTANTS: e = 1.6x 10"9C £0 = 8.85 x10_nF/m “o = 411x io‘7HJm Part A [6] (ii) Page 2 of I?
Question 1: Electricity A non conducting cylindrical pipe made of material with a relative
dielectric constant of one has an inside diameter of 2cm and an out
sf 1e diameter of 3cm. The axis of the pipe is aligned with the zaxis.
as shown. The height of the pipe is 4m. A charge ofSmC has been
deposited uniformly throughout the volume of the pipe (that is.
between 2 cm and 3 cm diameters).. [2] (i) Find the charge density in the cylinder p. ”2 Cm—o 0—3 cm—» The axis of the pipe is aligned with the z~axis of the coordinate system and the centre of the pipe is _)
at the origin, as shown. Find the electric ﬁeld along the xaxis, E(x.0,0) , as a function of the
distance from the z—axis for 0<x<6cm. )
For 0<x<1crn E(x,0,0) = _)
For 1<x<l.5cm E(x.0,0)= .9
For l.5<x<6cm E(x,D,O) = Page 3 of 12 Question 1: Electricity
Part B Two parallel conducting plates of area 0.5m2 each
Curiaiicn'nt: Plum. 0“ are 3cm apart. One plate (B) lies in the xy plane,
' while the other (A) intersects the positive zaxis.
The gap between the plates extends from 2:0 to
NWCE‘im‘mK z=3cm. Between the plates and parallel to the
0 plates lies a non conducting slab of thickness 1cm
and relative dielectric constant of one. It is posi 1‘5 cm tioned between 2:0.5cm and 1:1.5cm, as shown
in the ﬁgure. Ignore all edge effects, that is, treat l.0 cm y the electric ﬁeld distribution as if the plates and
slab were of inﬁnitely large area. 0.5 cm Conducting Plate. 03
I
[3] (i) Find the capacitance of the assembly. Now. a total charge of lSmC is uniformly distributed throughout the volume of the non conducting slab. [4] (ii) Find the surface charge densities 0A and 03 on the inner surfaces of plates A and B, respectively. A voltage source is used to force VAB=5V. Assume this does not affect the distribution of the lSmC charge
in the slab. [5] (iii) Find the electric ﬁeld between the conducting plates. For 0<z<0.5cm 312) = For l.5<z<3cm Eu) : Page 4 of I? Question 2: Magnetism A 15cm long cylindrical wire has a crosssection of 3mm2 and _ . . . —8
Loop m {1 reststiwly of 2 x 10 9cm.
[1] (i) What is the resistance of the wire. )
B
_—p—
l’ Lt“ k Loop :11 r=n The wire is bent into a circular loop, the ends connected, and placed in the xyplane. with the centre of the loop at the origin.
Loop :tt r=r The loop is made to perform very small sinusoidal oscillations
about the xaxis. as shown in the ﬁgure. The angle a = 0.05 sin( lOt rad/s) rad. A uniform magnetic ﬁeld
.) a
B 2 2j T induces a current, [(t), in the loop. Him: (a) ¢B(t = O) = O, (b) sinaza for small a.
[6] (ii) Find the magnetic flux through the loop, ¢B(t) . as a function of time. Loop at r=U [3] (iii) Find the current in the loop, i(t). as a function of time. Page 5 of I2 Question 2: Magnetism [4] (it!) Find the magnitude of the current at time I=O.3s. Indi
cate the direction of the current in the ﬁgure on the left. Loop at 1:0 [6] (v) Assuming that the induced current in the loop is i(t) = A sin(u)t). ﬁnd the total magnetic ﬁeld at the h)
center of the loop BToutli) as a function of time. Give the three components. Page 6 of 12 Question 3: DC Circuits Part A
[7] (i) Consider the circuit shown below. Assuming that RL is a load resistor, ﬁnd the Norton equivalent for the rest of the circuit (excluding RL). Sketch the Norton equivalent circuit and give the values
of all of its parameters. VS=10V Norton equivalent circuit i
[4] (ii) Using the circuit parameters from part m above, ﬁnd the value of the load resistor RI, that
maximizes the power transfer to the load and ﬁnd this maximum power. Page '1' of l2 Question 3: DC Circuits Part B
Consider the circuit shown below [2] (i) Label all of the currents in the ﬁgure above and write the equations required to ﬁnd the voltages V].
V2, V3 and V4. using nodal analysis. Take V4 as the reference node. [7] (ii) You know from direct measurements the following:
(a) For Vs=10 V and RL=10 kQ you measure V2\’4=4.6V and V3V4=3.8V,
(b) For V5=5 V and RL=I k9 you measure V2'V4=2.2V and V3V4=2.IV.
Using the nodal analysis approach find the values of R2 and R4. Page 8 01'12 Question 4: Transient Analysis The switch is closed at time t=O in the circuit shown below and then opened again at time t=2ms. Assume
that the switch was open for a long time prior to time zero. 1:0? IOV llLF 2kQ 21(3) [2] (i) Find the value of vC(l) for t = 0 and t = 0+. [5] (ii) Write the differential equation for vdt) as a function of time for O < t < 2ms. Page 9 of l2 Question 4: 'Iiansient Analysis [4] (iii) Find the analytical expression for vdt) as a function of time for 0 <.' t < 2ms; give the numerical
values of all of the parameters. Find the value of vC(t) at time t=2ms. vdt): vdt=2ms)= [5] (iv) Find the analytical expression for vdt) as a function of time for t > Ems; give the numerical values
of all of the parameters. vdt) = Page100f12 Question 4: Transient Analysis [4] (v) Sketch lid!) for time 2ms < t < 5ms; clearly label the axes and give the numerical values of all
critical parameters. Page Hofl‘.J Question 5: AC Circuits Part A
[10] Consider the RLC circuit shown below. The AC source v5(t) is lOOVmS at o) =1000rad/s. Assume that the value of the variable capacitor is set to 0.1 mF and that the phase of the voltage source at time t=0 is
(punt/4. Find the current in the inductor and write it in phasor form. Give the value of the current in the inductor at time t=1ms and the power being delivered to the inductor at time [=1 ms. Page I? of IE Question 5: AC Circuits
Part B Consider the RLC circuit shown below (the same circuit as in Part A). The AC source v50) is 100%,“s at
to =1000radfs. [2] (i) Write the expressions for the source voltage in the time domain and in the frequency domain [8] (ii) Find the speciﬁc value of the variable capacitor for which the current in the source is maximum
and give the expression for this current in the time domain. ...
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 Winter '08
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