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Unformatted text preview: EXQW G)me Mam [AWLM/wmg University of Toronto
Faculty of Applied Science and Engineering ECE 1598 — Fundamentals of Electricity and Electric Circuits
Final Examination
April 2004 Examiners: Professors M. Joy and S. Zukotynski Name:
Student no: Print your name and student number at the top of each page Parts 1 and 2 consist of three questions, parts 3 through 6 of two questions.
Each question is worth 5 marks. Physical constants: Elementary charge = 1.6x10_19C
Permittivity constant = 8.85x10'l2 F/m
Permeability constant : 1.26>~<10’6 H/m Nonprogrammable calculator allowed Place your ﬁnal answer in the box Use the back of the previous page for rough work
ALL WORK TO BE MARKED IS TO BE DONE ON THESE SHEETS There are two spare pages at the back Page I of J? Name: Student no: Part 1  Electric Fields — Question 1 Four stationary point charges. Q1, Q2. Q3, and Q4. of strength Q, —Q, Q, and —Q and are
located at points (—Zr, 0, 0). (—r, O, 0), (r, 0. 0), and (Zr. 0. 0), respectively. :1) Provide expressions for the electric field EU, y, 2) due to each charge, and the total
electric ﬁeld, for an arbitrary point in space b) Suppose a test charge q5 = —Q is placed at location (2r, —2r, 0). Draw a clear sketch
of the charge system and the test charge in the plane 2 = 0, depicting the forces on the
test charge due to each charge in the system, and the net force 0n the test charge. .1 c) Provide sketches (3 total) of the components of 50:, 0, O)as a function of it, clearly labelling all salient features. T— _________d_r___’_ Pace 2 of l7 Name: Student no.2 Part 1 — Electric Fields — Question 2 a) A charge of Q =x/5HC is located at the centre ofa regular tetrahedron of side 1mm,
that is, the charge is equally distant from the four vertexes of the tetrahedron. Find the
flux of the electric ﬁeld through one face of the tetrahedron. (Note: a tetrahedron is a pyramid built using four identical regular triangles.) b) An inﬁnitely long straight rod of radius 10cm is positioned so that the axis of the rod
coincides with the xaxis. The rod is made of insulating material of dielectric
coefﬁcient 8 = 280. The rod carries a charge uniformly distributed throughout its
volume of volume charge density p = lpC/m'. Find the electric ﬁeld at the point with
coordinates (0, 2cm, 0). Pace 3 of 17' Name: Student no: Part 1 —~ Electric Fields — Question 3 Consider two inﬁnite sheets of charge and a point charge. One sheet coincides with the x
= 0 plane and carries a uniform charge of surface charge density 0 = +lOuC/m2. The
second coincides with the y = 0 plane and carries a uniform charge of surface charge
density 0 = ~5uC/mz. A point charge of luC is located at (5cm. 5cm. 0). Assume the potential at the point (lOcrn. 10cm, 0) is zero and find the potential V at the
point (2cm, 2cm. 0}. a) Due to the Hggml sheet of charge; V(lOt1C/m2)= b) Due to the —5 lm2 sheet of charge; WSpLC/rn2) = c) Due the point charge. V(1HC)= d) The total potential at the point (2cm, 2cm. 0). V(Total) = I U I B Page 4 of 17 Name: Student n0.: Part 2  Magnetic Fields  Question 1 A square loop of side L = 20cm carries a current of
I = 500mA as shewn in the diagram. The loop is in
the XY plane and centered on the origin. This problem concerns the magnetic ﬁeld if at the origin
caused by the current I. a) What is the direction of B ? b) Derive a symbolic expression for the magnitude of E.
xdx 1 J dx x HINTS: I—*—————= W‘W; 2 2"2 2 2’2;
x+a) x+a c) What is the numerical value of the magnitude of I} ? Page 5 of I? Name: Student no.2 Part 2  Magnetic Fields  Question 2 A parallel plate capacitor (in vacuum) is placed in a
uniform magnetic ﬁeld of strength 0.1T. The direction of the magnetic ﬁeld is parallel to the plates as indicated in the ﬁgure. The capacitor is charged to pr0vide a uniform
electric ﬁeld of 105Wm in the direction shown. An electron enters the capacitor from the
left with a speed of 3x106m/sec; its direction of motion is at right angles to both the
magnetic ﬁeld and the electric ﬁeld as shown. a) What is the net force in Newtons (magnitude and direction) that the electron
experiences as it enters the capacitor? b) If we could change the strength of the magnetic field, would it then be possible for
this electron to pass through the capacitor in a straight line (undeﬂected)? Circle the correct answer.
YES NO c) If your answer is yes give the strength of the magnetic ﬁeld. If your answer is no, but
there is a different direction of the magnetic ﬁeld that would allow the electron to
pass through the capacitor in a straight line, give the required direction and the
strength of the magnetic ﬁeld. If your answer is no and there is no magnetic ﬁeld that
would allow the electron to pass in a straight line, then write "No magnetic ﬁeld" in
the answer box. The direction is: Page 6 of I? Name:_ Student no.2 Part 2 — Magnetism — Question 3 3' A uniform magnetic ﬁeld of strength 3 exists in the —k
direction. A railing is bent into a
parabola described by the
equation y = (x—a)2 b as
shown in the ﬁgure and is placed
on the xy plane. A rolling rod
of cross section radius r and
resistivity p moves at a constant velocity u in the direction. The rod is at y = 0 at time zero.
Assume that the rail has
negligible resistance and the rod
makes intimate electrical contact
with the rail. 7. a) Find the magnetic flux through the closed area formed by the rod and the railing. (I) b) Find the induced current. c) What is the induced current if the railing is place on the yz plane? Page 7' of 17 Name: Student no.: Part 3 — DC Circuits  Question 1 Consider the circuit in the ﬁgure shown below 3) Find the Thevenin equivalent with respect to the load RL. terminals (2!) in the circuit.
Sketch the circuit and give the values of VT}, and Rm. Thévenin equivalent circuit b) In terms of V and R, what is the value of RL for maximum power transfer to RI, and
what is this maximum power? Page 8 0f l7 Name: Student no.: Part 3 — DC Circuits — Question 2 3) Assume that the circuit has been connected for a long time and has reached its steady
state. Use nodal analysis to find the potential of nodes A and B with reSpect to
ground. 2k 2V b) If the ground reference was moved to node A as shown below, what would be the
potential of node B with respect to ground. ‘ f1nF Bk 1mA Page 9 of l? Name:__ Student no.: Part 4 — OpAmps — Question 1 In the circuit shown below, assume that the OpAmp is ideal. IOKQ (i) Find the voltage gain, Vol V; and the output current, in. (ii) Find the input resistance, R. the power gain. Pol Pi, and the total power delivered by
the opamp, PT. Page IO of]? Name: Student no.2 Part4 — OpAmps — Question 2 In the circuit shown below, assume that the OpAmp is ideal. The input signal vi“)
increases linearly for 0<r<0.ls. as shown in the graph. Otherwise. v.(r) =0. CzlouF 0 0.1 5 Find the output voltage, 1:00) for —ls<!<+ls and sketch 120(1). VA!) = 1W) Page ll 0f 17 Name: Student no.1 Part 5  Transients  Question 1 Consider the circuit below. Assume that Vc(t = 0) = O, R = 10 k9, C = 100 ,uF and that
the strength of the current source varies with time as given below. 0 t<Os
i(t)= {[mA] 0<t<ls
1[mA] t>1s Find 126(1) for all I. Also Sketch v60) vs. I. for t<05z
125(1): for O<t<ls: for t>lsr VCU) Page 12 of I? Name: _Stuclent not: Part 5 — Transients — Question 2 At time t=0, both switches in the following circuit are thrown. 12 mH a) Before (:0, what are the voltages V3,: and V1??? b) Just after [=0, what are the voltages V2; and V32? (3) What will VR; and V9; be at t : co? (1) Calculate the time constants, rm and on that will govern the behaviour of the voltages
uR,(t) and 1232(1), reSpectively. c) Write the equations for vmm and mum for 1>0. Page 13 of I? Name: Student no.1 Part 6 — AC Circuits — Question 1 Consider the circuit shown below. a) Find the effective impedance Z4; seen by the source. [3) Find the power factor pF for power delivered to this impedance. pF= c) Find the current phasor IX d) Find the voltage phasor. VI Page 14 of 17 Name: Student no: Part 6 — AC Circuits — Question 2 3) Assume that the frequency of the source is 100112 and that each of the impedances
represents a single component (R. C. or L). Find the components represented by all of
the impedances. b Give V5 in the time domain VF“): aﬁ/Give VI in the time domain (Use the result from Question1(d)). V.(t)= <11) Specify if V, is lagging or leading V5 and determine the time delay. Leading or Lagging Page 15 of 17' Name: Student no.: THIS PAGE WAS 1.315131~ BLANK INTENTIONALLY. USE IT FOR ROUGH WORK. Page 16 of 17 Name: Student no; THIS PAGE WAS LEFT BLANK INTENTIONALLY. USE IT FOR ROUGH WORK. Page 17' of 17 ...
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