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无代写-ELEC104P
时间：2022-04-26
ELEC104P
Page 1 of 7 TURN OVER
Physical Constants:
Electron Charge, e = 1.6 × 10-19 C
Boltzmann constant = 1.38 × 10
−23 J∙K-1
= / = 25 mV at room temperature
Speed of light = 3 × 108 m∙s−1
Planck’s constant ℎ = 6.63 × 10−34 J∙s−1
o = 8.85 × 10-12 F∙m-1
o = 4 × 10-7 H∙m-1
Answer ALL the questions in Section A. Answer TWO questions from Section B
There is a formula sheet on page 7.
SECTION A (There are EIGHT questions in this section)
1. Sketch a diagram to show the direction of the cross product of the two vectors (1,1,0)
and (-1,1,0) and state the magnitude of the cross product. [6 marks]
2. A point charge, q, is placed at the point (2,6,2). What is the electric field due to this
charge at the point (-1,2,-3)? [6 marks]
3. By using Gauss’ law (electric), equation [6] on the formula sheet, derive the
conditions on the normal (perpendicular) component of the electric flux density, D,
at the surface of a conductor with a surface charge density of σ C∙m-2. Explain the
steps in your answer. [7 marks]
4. (a) Sketch the magnetic flux around a straight current carrying wire.
(b) A positive charge is moving parallel to the wire, in the same direction as the
current, what is the direction of the force on the charge? Justify your answer.
[6 marks]
5. Ampère’s law is given by =
Sc
dd SJcH and can be used to find the magnitude
of the magnetic field, H, inside a current carrying wire (current density J), at a
distance r from the centre of the wire ==
Sc
drHd SJcH 2
A wire of radius 10 mm carries a uniformly distributed current of 1 A, what is the
magnetic field, H, at a distance of 4 mm from the centre of the wire?
[6 marks]
ELEC104P
Page 2 of 7 CONTINUED
6. Explain the formation of eddy currents in a conductor. Give an example of their
importance in engineering. [6 marks]
7. Briefly explain the origin of the depletion region in a pn junction. A silicon pn
junction is formed at the interface between boron-doped p-type silicon and arsenic-
doped n-type silicon. Figure 7.1 shows a sketch of the charge density across the pn
junction. If the arsenic concentration in the n-type silicon is 1016 /cm3, deduce the
boron concentration in the p-type silicon.
[6 marks]
Figure 7.1
8. A square solar cell of length 10 cm has an open-circuit voltage (VOC) of 0.6 V and
short-circuit current (ISC) of 25 mA/cm
2 under direct sunlight. Sketch the current-
voltage relationship of the solar cell, both i) in the dark, and ii) under direct
sunlight, taking care to label relevant values. Estimate the maximum power which
can be extracted from such a solar cell.
[7 marks]
ELEC104P
Page 3 of 7 TURN OVER
SECTION B Answer TWO out of FOUR questions from this section.
9. (a) State Gauss’ law (electric) in words. [4 marks]
(b) The electric field at a distance R from a line charge, Q C∙m-1, is given by
R
r R
Q
aE
02
= , where Ra is a unit vector in the radial direction. Using this,
obtain an expression for the capacitance of a coaxial cable with an inner conductor
of radius a, and an outer conductor of radius b, filled with a dielectric of relative
dielectric constant r.
[8 marks]
(c) A section through a coaxial cable is drawn as two concentric circles, one
representing the inner conductor and the other the outer conductor.
(i) Sketch a section through a coaxial cable and show the electric
field when there is a positive charge (+Q) on the inner conductor
and a negative charge (-Q) on the outer conductor.
(ii) Sketch a second diagram showing the same section, but now
show the magnetic field when a current (I) flows in one
direction in the inner conductor and the same size current flows
in the opposite direction in the outer conductor.
(iii) What are the fields outside of the coaxial cable and why is this
important?
[8 marks]
(d) Stray capacitance is an important consideration in integrated circuits. Why?
[5 marks]
ELEC104P
Page 4 of 7 CONTINUED
10. (a) The electric field in the radial direction, ER at a distance R and angle from the axis
of a dipole, of two charges, +Q and – Q separated by l, shown in figure 10.1, is given
by:
cos
R
Ql
ER 34
=
Figure 10.1
The definition of the potential difference BA VV − between two points A and B is
given by equation [3] on the formula sheet. For a dipole where Q = 2 × 10-19 C and
l = 10 nm, what it the potential difference between two points r1= 100 nm and
r2 = 200 nm at the same angle
45= to the dipole axis?
[9 marks]
(b) (i) Using the electric field at a distance R from a point charge,
24
R
o r
q
R
=E a , derive the potential at a distance r away from a point
charge Q.
(ii) A point charge of 10 µC is surrounded by a spherical conductor of inner
radius r = 5 cm and outer radius r = 6 cm; the sphere is filled with a
dielectric of εr = 2.5. Sketch a plot of the potential from r = 1 cm to
r = 10cm.
(iii) What will change if the conductor is connected to ground?
[12 marks]
(c) Electrons in a conductor are accelerated when a potential is applied to the
conductor. How is this consistent with Ohm’s law, which predicts that the current is
constant in a conductor, i.e. the electrons travel with an average constant velocity?
[4 marks]
E
R
r
1
r
2
q
E
θ
+Q -Q
ELEC104P
Page 5 of 7 TURN OVER
11. (a) Using Faraday’s law, describe how a transformer, shown in figure 11.1, can be used
to convert an a.c. voltage of amplitude v1 to amplitude v2, and derive the
dependence on the number of turns of the coils N1 and N2. Include the assumptions
that you are making in your answer.
[8 marks]
Figure 11.1
(b) The supply, S, shown in figure 11.1, has internal resistance R. The coil N1 has
resistance R1. The supply is switched on at t = 0 to give a constant voltage V. At a
short time later, t = T, a load RL is switched across the second coil. Sketch or
describe the current that flows through the coil N1 as a function of time and explain
your answer.
[7 marks]
(c) What is the difference between a hard and soft magnetic material? Give an example
of an application for each. [6 marks]
(d) Charged particles are emitted by the sun during a ‘solar storm’. They are highly
energetic and some will reach the earth. What role does the earth’s magnetic field
have in protecting life from these energetic particles?
[4 marks]
i
1
N
1
N
2
v2 S v1
ELEC104P
Page 6 of 7 CONTINUED
12. (a) Sketch a cross-section of an n-channel MOSFET, labelling all relevant parts.
[4 marks]
(b) State the condition under which an n-channel MOSFET enters the saturation regime,
in terms of the drain-source voltage VDS, the gate voltage VGS and the threshold
voltage Vthr. Briefly describe the physical origin of saturation in an n-channel
MOSFET, making use of a diagram as necessary, and use it to justify the condition
for saturation. [5 marks]
(c) An n-channel MOSFET (transconductance parameter gn = 40 mA/V2 and Vthr = 0.75
V) is used in the circuit shown in Figure 12.1. An input signal Vi is applied to the
gate, and an output signal Vo is measured on the drain.
(i) Consider first that Vi is a 1 V DC signal. Making an assumption about the
regime of operation of the MOSFET, calculate a value for Vo, assuming it is
measured using a high-impedance voltmeter. Use your answer to justify your
assumption.
[8 marks]
(ii) Now consider that Vi contains an additional 20 mV (peak-to-peak) AC signal
superimposed on the 1 V DC signal. What is the resulting (peak-to-peak)
amplitude of the AC signal on Vo ?
[4 marks]
(iii) State and explain how the following physical dimensions of the MOSFET
could be changed in order to increase the gain of this amplifier circuit: (A)
oxide thickness tox, (B) gate width W, (C) gate length L.
[4 marks]
Figure 12.1
ELEC104P
Page 7 of 7 END OF PAPER
FORMULA SHEET
The symbols have the usual meaning.
[1] Permittivity D = 0E + P = 0rE
[2] Permeability B = μr μo H and B = 0 (H + M)
[3] Electric potential between A and B −=−
A
B
BA dVV cE
[4] Ohm’s law J=E
[5] Gauss’ law (magnetic) 0=
S
dSB
[6] Gauss’ law (electric) =
VS
dvd SD
[7] Continuity of charge −=
+ V'SS
dv
dt
d
d SJ
[8] From the Biot-Savart law
=
2
0
4 r
I
d r
udl
B
[9] The Lorentz force: BvEF += QQ
[10] Maxwell via Ampère’s law: +=
SSc
d
dt
d
dd SDSJcH
[11] Faraday’s law −=
Sc
M d
dt
d
d SBcE
Page 1 of 7 TURN OVER
Physical Constants:
Electron Charge, e = 1.6 × 10-19 C
Boltzmann constant = 1.38 × 10
−23 J∙K-1
= / = 25 mV at room temperature
Speed of light = 3 × 108 m∙s−1
Planck’s constant ℎ = 6.63 × 10−34 J∙s−1
o = 8.85 × 10-12 F∙m-1
o = 4 × 10-7 H∙m-1
Answer ALL the questions in Section A. Answer TWO questions from Section B
There is a formula sheet on page 7.
SECTION A (There are EIGHT questions in this section)
1. Sketch a diagram to show the direction of the cross product of the two vectors (1,1,0)
and (-1,1,0) and state the magnitude of the cross product. [6 marks]
2. A point charge, q, is placed at the point (2,6,2). What is the electric field due to this
charge at the point (-1,2,-3)? [6 marks]
3. By using Gauss’ law (electric), equation [6] on the formula sheet, derive the
conditions on the normal (perpendicular) component of the electric flux density, D,
at the surface of a conductor with a surface charge density of σ C∙m-2. Explain the
steps in your answer. [7 marks]
4. (a) Sketch the magnetic flux around a straight current carrying wire.
(b) A positive charge is moving parallel to the wire, in the same direction as the
current, what is the direction of the force on the charge? Justify your answer.
[6 marks]
5. Ampère’s law is given by =
Sc
dd SJcH and can be used to find the magnitude
of the magnetic field, H, inside a current carrying wire (current density J), at a
distance r from the centre of the wire ==
Sc
drHd SJcH 2
A wire of radius 10 mm carries a uniformly distributed current of 1 A, what is the
magnetic field, H, at a distance of 4 mm from the centre of the wire?
[6 marks]
ELEC104P
Page 2 of 7 CONTINUED
6. Explain the formation of eddy currents in a conductor. Give an example of their
importance in engineering. [6 marks]
7. Briefly explain the origin of the depletion region in a pn junction. A silicon pn
junction is formed at the interface between boron-doped p-type silicon and arsenic-
doped n-type silicon. Figure 7.1 shows a sketch of the charge density across the pn
junction. If the arsenic concentration in the n-type silicon is 1016 /cm3, deduce the
boron concentration in the p-type silicon.
[6 marks]
Figure 7.1
8. A square solar cell of length 10 cm has an open-circuit voltage (VOC) of 0.6 V and
short-circuit current (ISC) of 25 mA/cm
2 under direct sunlight. Sketch the current-
voltage relationship of the solar cell, both i) in the dark, and ii) under direct
sunlight, taking care to label relevant values. Estimate the maximum power which
can be extracted from such a solar cell.
[7 marks]
ELEC104P
Page 3 of 7 TURN OVER
SECTION B Answer TWO out of FOUR questions from this section.
9. (a) State Gauss’ law (electric) in words. [4 marks]
(b) The electric field at a distance R from a line charge, Q C∙m-1, is given by
R
r R
Q
aE
02
= , where Ra is a unit vector in the radial direction. Using this,
obtain an expression for the capacitance of a coaxial cable with an inner conductor
of radius a, and an outer conductor of radius b, filled with a dielectric of relative
dielectric constant r.
[8 marks]
(c) A section through a coaxial cable is drawn as two concentric circles, one
representing the inner conductor and the other the outer conductor.
(i) Sketch a section through a coaxial cable and show the electric
field when there is a positive charge (+Q) on the inner conductor
and a negative charge (-Q) on the outer conductor.
(ii) Sketch a second diagram showing the same section, but now
show the magnetic field when a current (I) flows in one
direction in the inner conductor and the same size current flows
in the opposite direction in the outer conductor.
(iii) What are the fields outside of the coaxial cable and why is this
important?
[8 marks]
(d) Stray capacitance is an important consideration in integrated circuits. Why?
[5 marks]
ELEC104P
Page 4 of 7 CONTINUED
10. (a) The electric field in the radial direction, ER at a distance R and angle from the axis
of a dipole, of two charges, +Q and – Q separated by l, shown in figure 10.1, is given
by:
cos
R
Ql
ER 34
=
Figure 10.1
The definition of the potential difference BA VV − between two points A and B is
given by equation [3] on the formula sheet. For a dipole where Q = 2 × 10-19 C and
l = 10 nm, what it the potential difference between two points r1= 100 nm and
r2 = 200 nm at the same angle
45= to the dipole axis?
[9 marks]
(b) (i) Using the electric field at a distance R from a point charge,
24
R
o r
q
R
=E a , derive the potential at a distance r away from a point
charge Q.
(ii) A point charge of 10 µC is surrounded by a spherical conductor of inner
radius r = 5 cm and outer radius r = 6 cm; the sphere is filled with a
dielectric of εr = 2.5. Sketch a plot of the potential from r = 1 cm to
r = 10cm.
(iii) What will change if the conductor is connected to ground?
[12 marks]
(c) Electrons in a conductor are accelerated when a potential is applied to the
conductor. How is this consistent with Ohm’s law, which predicts that the current is
constant in a conductor, i.e. the electrons travel with an average constant velocity?
[4 marks]
E
R
r
1
r
2
q
E
θ
+Q -Q
ELEC104P
Page 5 of 7 TURN OVER
11. (a) Using Faraday’s law, describe how a transformer, shown in figure 11.1, can be used
to convert an a.c. voltage of amplitude v1 to amplitude v2, and derive the
dependence on the number of turns of the coils N1 and N2. Include the assumptions
that you are making in your answer.
[8 marks]
Figure 11.1
(b) The supply, S, shown in figure 11.1, has internal resistance R. The coil N1 has
resistance R1. The supply is switched on at t = 0 to give a constant voltage V. At a
short time later, t = T, a load RL is switched across the second coil. Sketch or
describe the current that flows through the coil N1 as a function of time and explain
your answer.
[7 marks]
(c) What is the difference between a hard and soft magnetic material? Give an example
of an application for each. [6 marks]
(d) Charged particles are emitted by the sun during a ‘solar storm’. They are highly
energetic and some will reach the earth. What role does the earth’s magnetic field
have in protecting life from these energetic particles?
[4 marks]
i
1
N
1
N
2
v2 S v1
ELEC104P
Page 6 of 7 CONTINUED
12. (a) Sketch a cross-section of an n-channel MOSFET, labelling all relevant parts.
[4 marks]
(b) State the condition under which an n-channel MOSFET enters the saturation regime,
in terms of the drain-source voltage VDS, the gate voltage VGS and the threshold
voltage Vthr. Briefly describe the physical origin of saturation in an n-channel
MOSFET, making use of a diagram as necessary, and use it to justify the condition
for saturation. [5 marks]
(c) An n-channel MOSFET (transconductance parameter gn = 40 mA/V2 and Vthr = 0.75
V) is used in the circuit shown in Figure 12.1. An input signal Vi is applied to the
gate, and an output signal Vo is measured on the drain.
(i) Consider first that Vi is a 1 V DC signal. Making an assumption about the
regime of operation of the MOSFET, calculate a value for Vo, assuming it is
measured using a high-impedance voltmeter. Use your answer to justify your
assumption.
[8 marks]
(ii) Now consider that Vi contains an additional 20 mV (peak-to-peak) AC signal
superimposed on the 1 V DC signal. What is the resulting (peak-to-peak)
amplitude of the AC signal on Vo ?
[4 marks]
(iii) State and explain how the following physical dimensions of the MOSFET
could be changed in order to increase the gain of this amplifier circuit: (A)
oxide thickness tox, (B) gate width W, (C) gate length L.
[4 marks]
Figure 12.1
ELEC104P
Page 7 of 7 END OF PAPER
FORMULA SHEET
The symbols have the usual meaning.
[1] Permittivity D = 0E + P = 0rE
[2] Permeability B = μr μo H and B = 0 (H + M)
[3] Electric potential between A and B −=−
A
B
BA dVV cE
[4] Ohm’s law J=E
[5] Gauss’ law (magnetic) 0=
S
dSB
[6] Gauss’ law (electric) =
VS
dvd SD
[7] Continuity of charge −=
+ V'SS
dv
dt
d
d SJ
[8] From the Biot-Savart law
=
2
0
4 r
I
d r
udl
B
[9] The Lorentz force: BvEF += QQ
[10] Maxwell via Ampère’s law: +=
SSc
d
dt
d
dd SDSJcH
[11] Faraday’s law −=
Sc
M d
dt
d
d SBcE
