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电子工程代写-ENG 3FK4

时间：2020-12-12

Student Name:____________________________________ Student Number________________________________

Page 1 of 4

ELEC ENG 3FK4

DAY CLASS Dr T R FIELD

DURATION: 2.5 HOURS

MCMASTER UNIVERSITY FINAL EXAMINATION DECEMBER 2019

THIS EXAMINATION PAPER INCLUDES 4 PAGES (includes the cover page) AND 6 QUESTIONS. YOU ARE

RESPONSIBLE FOR ENSURING THAT YOUR COPY OF THE PAPER IS COMPLETE. BRING ANY

DISCREPANCY TO THE ATTENTION OF YOUR INVIGILATOR.

Special

Instructions:

CLOSED BOOK

8.5”x11” crib sheet one side is allowed

Use of any calculator is allowed

Answer any FOUR questions

All questions carry equal weight

Explain all reasoning carefully

1. Consider a static volume current density )J(r' where 'r is the position vector of a point in the current

distribution. Show that the field generated at a point with position vector r , according to the Biot-Savart law,

vd

R

3

3

)'(

4

)(

RrJ

rB

,

in which 'rrR and ||RR , satisfies Maxwell’s magnetostatic equation JB ( should be

considered as constant).

Consider the magnetic vector potential defined by

vdR

3

4

J

A

.

Show that this potential satisfies the equation AB and the Lorenz gauge condition 0 A .

[Hint: see Useful formulae and apply the equation of continuity and anti-symmetry under interchange of ',rr ,

where appropriate.]

Student Name:____________________________________ Student Number________________________________

Page 2 of 4

2. Explain the meaning of dipole moment for a pair of equal and opposite charges q separated by a

displacement vector d , and the limiting case of a perfect dipole with moment p .

State the expression for the electrostatic potential for a single point charge q situated at 0r . By considering a

pair of equal and opposite charges in superposition, derive the following expression for the electrostatic

potential for a perfect dipole situated at the origin:

3

04

1

r

V

rp

.

where ),,( zyxr . [Hint: use the binomial approximation

2

12/1 1)1( for small values of , and the

cosine rule (see Useful formulae) applied to a triangle with vertices situated at the two charges and the field

point.]

Suppose that a dielectric material consists of dipoles with dipole moment P per unit volume (the polarization

density). Write down an expression for the contribution to the potential at a general point due to a volume

element '

3vd of the material in terms of the vector 'rrR where r , 'r are the position vectors of the field

point and material element respectively. Using the relation

3

1

'

RR

R

and the vector identity

fff '' ' AAA show that the potential due to the entire material occupying volume 'V with

boundary 'S is given by

'

'

'

'

4

1 3

'

2

'0

vd

R

Sd

R

V

V

pV

S

pS

where

pS' and pV' should be expressed in terms of the polarization density P , and whose physical

significance should be explained.

3. For a time dependent electromagnetic field show that, in the Lorenz gauge, its electric scalar and magnetic

vector potentials V and A satisfy the wave equations with source:

/

2

2

2

t

V

V , J

A

A

2

2

2

t

and identify the required Lorenz gauge. [You may assume Maxwell’s equations and the potential field relations

tV AE , AB .] Show that the retarded potentials

v vdR

V 3

4

][

, v vdR

3

4

][

J

A

are solutions to the above wave equations, where ][ denotes evaluation at the retarded time which should

be explained. [Hint: apply your argument to a general function f that satisfies the wave equation with

source g .]

Student Name:____________________________________ Student Number________________________________

Page 3 of 4

4. Given the total electromagnetic energy

1

2

W dv E D H B

show from Maxwell's equations that

( )

S V

W

d dv

t

E H S E J

Interpret this equation physically in terms of the Poynting vector and dissipation.

5. Starting from first principles, derive expressions for the Brewster angles pertaining to reflection of a plane

wave at oblique incidence, in the cases of parallel and perpendicular polarization. In this context, compare and

contrast the different polarizations for different types of electromagnetic media. [You may assume Snell's law

without proof.]

6. Explain the physical meaning of the reflection coefficient and standing wave ratio for electromagnetic

waves given, respectively, by the following expressions:

12

12

,

||1

||1

s .

Explain what is meant by the dominant mode for a rectangular waveguide. A rectangular waveguide with

cross sections shown in the figure below has dielectric discontinuity. Calculate the standing wave ratio

(using the above expressions) if the guide operates at 8 GHz in the dominant mode.

[Relevant physical constants: speed of light in vacuum

18 ms103 c ; intrinsic impedance of free space

Ohms 1200 .]

Student Name:____________________________________ Student Number________________________________

Page 4 of 4

Useful formulae

)B(A)A(BABBAB)(A

vv)(v 2

3

3

4 ( )

r

r

r

( ) ( ) ( ) A B B A A B

vv)(v 2

THE END

cosine rule:

Abccba cos2222

for triangle with side lengths cba ,, and

opposite angle A .

Page 1 of 4

ELEC ENG 3FK4

DAY CLASS Dr T R FIELD

DURATION: 2.5 HOURS

MCMASTER UNIVERSITY FINAL EXAMINATION DECEMBER 2019

THIS EXAMINATION PAPER INCLUDES 4 PAGES (includes the cover page) AND 6 QUESTIONS. YOU ARE

RESPONSIBLE FOR ENSURING THAT YOUR COPY OF THE PAPER IS COMPLETE. BRING ANY

DISCREPANCY TO THE ATTENTION OF YOUR INVIGILATOR.

Special

Instructions:

CLOSED BOOK

8.5”x11” crib sheet one side is allowed

Use of any calculator is allowed

Answer any FOUR questions

All questions carry equal weight

Explain all reasoning carefully

1. Consider a static volume current density )J(r' where 'r is the position vector of a point in the current

distribution. Show that the field generated at a point with position vector r , according to the Biot-Savart law,

vd

R

3

3

)'(

4

)(

RrJ

rB

,

in which 'rrR and ||RR , satisfies Maxwell’s magnetostatic equation JB ( should be

considered as constant).

Consider the magnetic vector potential defined by

vdR

3

4

J

A

.

Show that this potential satisfies the equation AB and the Lorenz gauge condition 0 A .

[Hint: see Useful formulae and apply the equation of continuity and anti-symmetry under interchange of ',rr ,

where appropriate.]

Student Name:____________________________________ Student Number________________________________

Page 2 of 4

2. Explain the meaning of dipole moment for a pair of equal and opposite charges q separated by a

displacement vector d , and the limiting case of a perfect dipole with moment p .

State the expression for the electrostatic potential for a single point charge q situated at 0r . By considering a

pair of equal and opposite charges in superposition, derive the following expression for the electrostatic

potential for a perfect dipole situated at the origin:

3

04

1

r

V

rp

.

where ),,( zyxr . [Hint: use the binomial approximation

2

12/1 1)1( for small values of , and the

cosine rule (see Useful formulae) applied to a triangle with vertices situated at the two charges and the field

point.]

Suppose that a dielectric material consists of dipoles with dipole moment P per unit volume (the polarization

density). Write down an expression for the contribution to the potential at a general point due to a volume

element '

3vd of the material in terms of the vector 'rrR where r , 'r are the position vectors of the field

point and material element respectively. Using the relation

3

1

'

RR

R

and the vector identity

fff '' ' AAA show that the potential due to the entire material occupying volume 'V with

boundary 'S is given by

'

'

'

'

4

1 3

'

2

'0

vd

R

Sd

R

V

V

pV

S

pS

where

pS' and pV' should be expressed in terms of the polarization density P , and whose physical

significance should be explained.

3. For a time dependent electromagnetic field show that, in the Lorenz gauge, its electric scalar and magnetic

vector potentials V and A satisfy the wave equations with source:

/

2

2

2

t

V

V , J

A

A

2

2

2

t

and identify the required Lorenz gauge. [You may assume Maxwell’s equations and the potential field relations

tV AE , AB .] Show that the retarded potentials

v vdR

V 3

4

][

, v vdR

3

4

][

J

A

are solutions to the above wave equations, where ][ denotes evaluation at the retarded time which should

be explained. [Hint: apply your argument to a general function f that satisfies the wave equation with

source g .]

Student Name:____________________________________ Student Number________________________________

Page 3 of 4

4. Given the total electromagnetic energy

1

2

W dv E D H B

show from Maxwell's equations that

( )

S V

W

d dv

t

E H S E J

Interpret this equation physically in terms of the Poynting vector and dissipation.

5. Starting from first principles, derive expressions for the Brewster angles pertaining to reflection of a plane

wave at oblique incidence, in the cases of parallel and perpendicular polarization. In this context, compare and

contrast the different polarizations for different types of electromagnetic media. [You may assume Snell's law

without proof.]

6. Explain the physical meaning of the reflection coefficient and standing wave ratio for electromagnetic

waves given, respectively, by the following expressions:

12

12

,

||1

||1

s .

Explain what is meant by the dominant mode for a rectangular waveguide. A rectangular waveguide with

cross sections shown in the figure below has dielectric discontinuity. Calculate the standing wave ratio

(using the above expressions) if the guide operates at 8 GHz in the dominant mode.

[Relevant physical constants: speed of light in vacuum

18 ms103 c ; intrinsic impedance of free space

Ohms 1200 .]

Student Name:____________________________________ Student Number________________________________

Page 4 of 4

Useful formulae

)B(A)A(BABBAB)(A

vv)(v 2

3

3

4 ( )

r

r

r

( ) ( ) ( ) A B B A A B

vv)(v 2

THE END

cosine rule:

Abccba cos2222

for triangle with side lengths cba ,, and

opposite angle A .