Download ECE221H1S Midterm Tuesday, Feb. 24, 2015

ECE221H1S
Midterm
LAST NAME:
Tuesday, Feb. 24, 2015
FIRST NAME:
Please place a check mark to indicate the tutorial section you are in.
TUT.
SEC.
101
102
103
104
105
106
TIME
TA
NAMES
GB404 Ali
GB412 Kenneth
GB304 Kenji
SF3201 Mehwish
SF3201 Jason
GB404 Min
ROOM
Thursday 9 – 11am
Thursday 9 – 11am
Friday 1 – 3pm
Friday 1 – 3pm
Friday 9 – 11am
Friday 9 – 11am
INSTRUCTIONS
1) Allowable aids: any non-programmable calculator, a double-sided handwritten
8.5”x11” aid sheet, and the Vector Calculus Aid Sheet (provided for you).
2) Show all your work.
3) Write clearly. You could lose marks if we cannot understand what you have written.
4) Make sure to take some time to think about the question before you answer it.
5) There are three questions, with the first one being entirely multiple choice. Make
sure to mark down all of your answers to Q1 on the provided Scantron Answer Sheet.
6) You may find some of the following equations of use:
cos sin
cos
= tan
= 0 sin
+ sin
=
1
(1 − cos 2 )
2
1
= (1 − cos 2 )
2
= cos + sin + 1 = 0 ( ) = 1
2
sin
=
( )
2
Q1: (25 marks)
Multiple Choice. Mark ALL of your answers to this question on the Scantron Answer Sheet provided.
You do not have to show your work for this question.
1. The electric scalar potential varies according to as
shown in the figure to the right. What conclusion
can be made about the electric field associated with
this scalar potential?
a) The electric field is zero for < 10 mm
b) The electric field is non-zero for > 20 mm
c) The electric field is zero for 10mm ≤ ≤
20mm
d) None of the above.
V
8V
∝
1
V=0
10 mm
20 mm
R
2. Consider Maxwell’s equations for electrostatics. Which statement about these equations is false?
a) Since the curl of the electrostatic field is always zero, it is possible to represent the electrostatic
field as the gradient of a scalar field.
b) The differential form of Gauss’s law is true at every point in space for all electric fields.
c) The integral form of Gauss’s law is true only if the charge distribution has the right type of
symmetry.
d) The integral form of Gauss’s law tells us that the net electric flux through any closed surface is
equal to the net charge inside that surface.
e) From Faraday’s law for electrostatic fields it can be proven that the line integral of the
electrostatic field is path independent.
3. The figure below shows a distribution of two point charges. Determine the direction of the electric
force on an electron if it was placed at the points P1, P2, and P3. Note, both P1 and P2 are located
along the mid-plane between the two charges.
P1
P2
+q
a)
b)
c)
d)
e)
P1: →
P1: ↑
P1: ←
P1: 0
P1: →
P2: 0
P2: →
P2: ←
P2: 0
P2: 0
P3: →
P3: 0
P3: →
P3: ←
P3: ←
P3
-q
4. A net charge of +Q is placed on a spherical metal shell as shown to the right.
Suppose a point charge of −10q is then placed in the center of the shell. What
is the charge density on the outer surface of the shell.
a)  s 
10q
4 b 2
b)  s 
Q
4 b 2
c)  s 
Q  10 q
4 b 2
d)  s 
10 q  Q
4 b 2
5. The following figure shows a conductive cylindrical shell of outer
radius c and inner radius b concentric with a conductive cylinder of
radius a. A charge densities of +ρs exists on the surface of the
conductive cylinder and an equal and opposite charge density of -ρs
exists on the inner surface of the cylindrical shell respectively.
Which of the following statement(s) is (are) true?
I. The potential difference between c and b is zero.
II. The potential difference between b and a is zero.
III. The potential difference between a point outside the cylinder and c is zero.
IV. The electric field between a point outside the cylinder and c is zero.
a)
b)
c)
d)
e)
I and IV are true
II and IV are true
I and III are true
II, III and IV are true
I, III, and IV are true
6. In the figure to the right, a situation is illustrated in which an
electric field impinges on a dielectric to dielectric interface.
From this picture what conclusion can be made? You may
assume that this situation accurately represents a correct electric
field configuration and that the vectors are drawn to their proper
scale.
a)
=
b)
>
c) There is no free charge density on the
dielectric/dielectric interface.
d) There is a positive free charge density
on the dielectric/dielectric interface.
e) There is a negative free charge density
on the dielectric/dielectric interface.
= 45°
7. Two dielectric materials are being placed within a two identical parallel-plate capacitors (side length
of the square plate is = 55cm) which both have charges of ± = ±4mC on their plates and
neither have a battery still attached. One material has a relative permittivity of
= 4, and the other
has a relative permittivity of
= 6, and the breakdown fields for the two dielectrics are given by
= 4 × 10 V/m and
= 2.7 × 10 V/m.
Under these conditions, would you have expected either or both of these dielectrics to fail? You may
assume that the size of the plates is much, much larger than the plate spacing.
a)
b)
c)
d)
e)
No, neither of them would fail.
Yes, both of them would fail.
Only material #1 would fail.
Only material #2 would fail.
It is not possible to determine this
without knowing the plate spacing, .
8. Given the fact that the electric flux density for a particular charge distribution is known to be:
0
< 2mm
= 3 R[C/m ] 2mm ≤ ≤ 4mm
0
> 4mm
Which statement below correctly describes the charge distribution that is responsible for this electric
field?
a) There is a uniform surface charge density at = 2mm.
b) There is a point charge at the origin ( = 0) and a volume
charge density given by
= 18 [C/m ] which exists only
for 2mm ≤ ≤ 4mm.
c) There is a uniform volume charge density given by
= 18 [C/m ] which exists only for 2mm ≤ ≤ 4mm.
d) There is a volume charge density given by
= 18 [C/m ]
which exists only for 2mm ≤ ≤ 4mm.
e) There is a volume charge density given by
= 6 [C/m ]
which exists only for 2mm ≤ ≤ 4mm.
f) There is a uniform surface charge density at = 2mm and a uniform
volume charge density which exists only for 2mm ≤ ≤ 4mm.
9. Three current-carrying wires are shown below and each wire has a current of
y
.
Variation of radius:
( )=
2
+
x
Case A
Length = l
Radius = a
Conductivity =
Case B
Length = l
Square side length = a
Conductivity = 2
Case C
Length = l
Start/End Radius = a
Middle Radius = 2a
Conductivity =
Consider the following statements. Determine which ones are true.
I. The largest current density is found in Case C.
II. The resistance of Case A is larger than that of Case B.
III. The resistance of Case C is smaller than that of Case A.
IV. The mobility of the material used in Case B is larger
than the mobility of the material used in Case C.
V. The mean free time of the material in Case C is larger
than that of Case A.
a)
b)
c)
d)
e)
f)
g)
I and IV are true
Only III is true
I, III, and IV are true
II, III and IV are true
II, III and V are true
III and IV are true
All statements are true
10. Consider the situation described in Q10 further. What should the conductivity in Case C be changed
to so that the resistance of Case B equals that of Case C (i.e.,
= )?
a)
= 0[S/m]
b)
= 2 c)
=
d)
=
e)
=3
f) No change is needed
Q2: (35 marks)
(18 marks) (a) An infinite line of charge with a uniform linear charge density + is positioned on the -axis
and exists in free space.
i) Apply Gauss’s Law to prove that the magnitude of the electric field intensity at a distance
from the -axis is given by: =
.
ii) Determine the magnitude of the potential difference between two points which are located at
P1(0, 0, z1) and P2(0, 0, z2) where z1<z2.
iii) Does the voltage increase or decrease in moving from P1 to P2?
Q2: (continued)
(17 marks) (b) A second infinite line of charge is added to the situation as described in part (a) above. This line
of charge has a uniform linear charge density of − is placed in the -plane with an orientation of
= 45° to the -axis.
i) Clearly draw the complete situation, including both lines of charge in the figure below.
ii) Determine the total electric field in vector form at P1(0, 2a, a).
iii) A Gaussian sphere of radius R is centred at the origin. Determine the total electric flux (Φ)
through this sphere.
Q3: (40 marks)
(20 marks) (a) A partial disk lies in the -plane and extends from = 0° to =
( < ), and is charged with a charge density of =
. Both
zero constants.
, and
and
= to =
are positive non-
i) Draw a picture of this situation.
ii) Find the total electric flux density at the origin.
iii) For what value of , with 0 <
≤ 2 , will the - and -components of the electric field
at the origin be equal and non-zero?
Q3: (continued)
(20 marks) (b) Two concentric metal spherical shells exist at = 1 cm and = 8 cm. A battery is connected
between them such that the shells are maintained with total charges of
= +4μC and
= −4μC. Between the shells two dielectric regions exist, with the first region from
1cm < < 5cm and the second region from 5cm < < 8cm. The first dielectric has a
relative permittivity of
= 4 and a conductivity of = 3 × 10 S/m and the second has a
relative permittivity of
= 6 and a conductivity of = 7 × 10 S/m.
For this problem you can assume that the system is operating in steady-state.
i) Find the resistance of the structure.
ii) Determine the total current passing through the sphere = 2 cm.
iii) Find the power loss in the first dielectric region.
iv) Find the power loss for the entire structure.
v) Use Gauss’s law and the electric field boundary conditions to determine the free surface
charge density which appears at = 5 cm. Note this is not necessarily zero.
Q3: (continued)
(b) (continued)