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SSuullaaiimmaannii UUnniivveerrssiittyy FFoouurrtthh SSttaaggee CCoolllleeggee ooff SScciieenncceess EElleeccttrroommaaggnneettiicc PPhhyyssiiccss DDeeppaarrttmmeenntt TTiimmee:: 7755 mmiinnuutteess

Q1/

transform into spherical coordinate and find its magnitude at point 3, 4, 0 . Q2/ Verify Stoke’s theorem for a vector field , in the segment of cylindrical surface

defined by 2 , , and 0 3 . Q3/ Determine the flux of through the entire prism shown in Fig(2). Q4/ Show that the scalar function , is harmonic at point √3, , 0 .

·

·

ρ

SSuullaaiimmaannii UUnniivveerrssiittyy -- CCoolllleeggee ooff SScciieenncceess -- PPhhyyssiiccss DDeeppaarrttmmeenntt

FFiirrsstt EExxaammiinnaattiioonn ((55//1111//22000099)) EElleeccttrroommaaggnneettiicc

Q1/

transform into spherical coordinate and find its magnitude at point 3, 4, 0 .

· · · · · · · · · · · ·

· · ·· · ·· · ·

00

0 1 0

√9 16 5

√25 5 05 90

43

53.13

5 sin 90 cos 53.13 5 0 cos 90 sin 53.13 5 cos 90 cos 53.13 5 0 sin 90 sin 53.13 .

3 0 0.018

Q2/ Verify Stoke’s theorem for a vector field , in the segment of cylindrical surface defined by 2 , 60 90 , and 0 3 .

The mathematical representation of Stokes’s theorem is given by:

dsAdlA ⋅×∇=⋅∫ ∫L L

rrr

The line integral around a closed path defined by these bounded region is as follows:

43

260coscos)ˆ(ˆ

0)ˆ(ˆ

0290coscosˆˆ

0ˆˆ

3

0

3

0

3

0

3

0

−=×−==−⋅=⋅

=−⋅=⋅

=×==⋅=⋅

=⋅=⋅

⋅+⋅+⋅+⋅=⋅

∫∫∫

∫∫

∫∫∫

∫∫

∫∫∫∫∫

zdzadza

ada

zdzadza

ada

zzz

a

d

zz

d

c

zzz

c

b

zz

b

a

a

d

d

c

c

b

b

aL

ρφ

φρ

ρφ

φρ

φ

φ

BdlB

BdlB

BdlB

BdlB

dlBdlBdlBdlBdlB

r

r

r

r

rrrrr

The left hand side of the Stokes’s theorem is : dsA ⋅×∇∫s

rr

[ ]ρφ

φρ

φρ

φφρ

ρφρ

ρφ

ρ

ρφ

φρ

ρ

ρ

aa

aaaz

aaa

z

z

ˆsinˆcos1

)00(ˆ)cos0(ˆ)0sin(ˆ1

cos00

ˆˆˆ1

2

2

−=×∇

⎥⎦

⎤⎢⎣

⎡−++++−=

∂∂

∂∂

∂∂

=×∇

B

B

rr

rr

Hence,

[ ]

43)()

210(

23)(

)03()3/cos2/(cos21)()(cos1)(

sinˆˆsinˆcos1)(

3

0

2/

3/

2/

3/

3

022

−=⋅×∇⇒−=⋅×∇

−×−×=××−=⋅×∇

−=⋅−=⋅×∇

∫∫

∫

∫ ∫∫∫

ss

s

ss

z

ddzaddzaa

dsBdsB

dsB

dsB

rrrr

rr

rr

ππφρ

φρρφφρφφ

ρπ

π

π

πρρφ

It is clearly seen that the left and right hand side has the same value, which indicates the validity of the Stokes’s theorem.

ρ

y

z

x

c

b

a

Q3/ Determine the flux of through the entire prism shown in Fig(2). The equation of surface (abc) is:

1

1 , 3 , 6

6 2 6 6 6 2

The equation of line (ab) is:

3 3 3 3 To find flux through the entire prism we can use divergence theorem:

. · · 3

. 3

. 3 6 6 2

. 3 6 3 3 6 3 3 23 3

2

. 27

. 2713

14

15 7.65

Q4/ Show that if the scalar function , is harmonic at point √3, , 0 .

H1r

∂∂r r

∂H∂r

1r sinθ

∂∂θ sinθ

∂H∂θ

1r sin θ

∂ H∂φ

H1r

∂∂r

r sin θ e1

r sinθ∂

∂θsinθ 2 sin θ cosθ e

1r sin θ

0

Hsin θr

r e 2r e2 e

r sinθsin θ 2 sin θ cos θ

Hsin θ

r e r 22 e

r 2 cos θ sin θ At point √3, , 0 , H 0 is not harmonic.