3-SEM-1-CBCS-PHY HC 1.pmd - ADP College

1. Answer the following questions : 1×7=7 (a) What is the geometrical interpretation of the scalar triple product of three vectors ?

Total number of printed pages–7

3 (Sem– 1/CBCS) PHY HC 1

2020 (Held in 2021)

PHYSICS (Honours)

Paper : PHY–HC–1016 ( Mathematical Physics–I )

Full Marks : 60

Time : Three hours

The figures in the margin indicate full marks for the questions.

(b) If

 

S

 

u du u d

R 

 , find



^ba R

 

u du .

(c) Find the Laplacian of the scaler field

3 2z y

x

 .

(d) Determine the order and degree of the differential equation

0

2 2 2

2  



 



 



 





dx x dy dx

y d

(e) What are the coordinate surfaces in orthogonal curvilinear coordinates ?

(f) Define Dirac delta function.

(g) Write the difference between Systematic error and Random error.

2. Answer any four of the following questions : 2×4=8 (a) If A

 

t

has a constant magnitude, then

show that dt

A d

is perpendicular to A .

(b) Prove that, the vector

kˆ y x jˆ z x iˆ z y

A 3 ^{4 2} 4 ^{3 2} 3 ^{2 2} is solenoidal.

(c) Show that



S

dS nˆ . A

, over any closed

surface S is equal to



R nˆ .kˆ dy nˆ dx

. A

,

where R is the projection of S on xy- plane.

(d) Solve the differential equation



y



dy



x



dx xy 1  ² 1 .

(e) State the transformation relation between the spherical polar co- ordinates



r,,



and Cartesian co- ordinates



x,y,z



. Obtain the volume elements in spherical polar co-ordinate.

3. Answer any three of the following

questions : 5×3=15

(a) How will you define divergence and curl of a vector _V^. Evaluate _^_.r^ and _^_r^.

(b) If A

is a vector, prove that



^{ A}



^



^.A



^A

 ₂

















 .

(c) Test the Exactness of the differential equation



^{5x 3x2y22xy3}



^{dx }



^{2x3y3x2y2 5y4}



^{dy 0}

and then solve it.

(d) Express ² in orthogonal curvilinear coordinates.

4. Answer any three of the following

questions : 10×3=30

(a) (i) Show that the surface integral of a vector _F^ and the volume integral of the divergence of the same vector obey the relation :

 





^{ }

V S

dV F . S

d .

F   

6

(ii) Evaluate



S

dS nˆ .

r , where S is a

closed surface. 4

OR

(b) Prove that ^

_ 

^



S C

S d . A d

.

A    

Š

^,

where C is the curve bounding the surface S. Hence find

Š

r.dr ^.

8+2=10

(c) Solve the following differential

equations : 5+5=10

(i)

 

^{xy cosx}

dx

x dy  

 2

1 ²

(ii) ²₂ 3 2y 0 dx

dy dx

y

d , subject to the

condition y

 

0 0 and y

 

0 1. (d) (i) Prove that spherical polar

coordinate system is orthogonal.

6 (ii) The probability density function of

a variable X is

P

 

X k k k k k k k X

13 11 9 7 5 3 :

6 5 4 3 2 1 0 :

Find ^P



^{X 4}



^{, P}



^{X 5}



^,



^{3X 6}



P . Here ^P

 

^{X is a}

probability density function. 4

(e) (i) Prove the expression



^^

 

x dx 1 where 

 

^{x 0 if}

0

x and 

 

^{x } if x 0. 5 (ii) Given the three vectors

jˆ iˆ C

kˆ jˆ B

kˆ jˆ iˆ A



















 2

Evaluate A



B C





 and show

that



^{B C}

 

^{B A.C}



^C



^A.B



A        









2+3=5

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