{Magå‘V {dÚwVJ{VH$s VWm gmno{ÎmH$Vm H$m {d{eîQ> {gÕmÝV Paper - MPH-04

MPH-04

June - Examination 2017

M.Sc. (Previous) Physics Examination Classical Electro Dynamics and

Special Theory of Relativity

{Magå‘V {dÚwVJ{VH$s VWm gmno{ÎmH$Vm H$m {d{eîQ> {gÕmÝV

Paper - MPH-04

Time : 3 Hours ] [ Max. Marks :- 80

Note: The question paper is divided into three sections. A, B and C. Write answer as per the given instructions. Check your paper code and paper title before starting the paper. Calculator are not allowed.

{ZX}e : ¶h àíZ nÌ "A', "~' Am¡a "g' VrZ IÊS>m| ‘| {d^m{OV h¡& à˶oH$

IÊS> Ho$ {ZX}emZwgma àíZm| Ho$ CÎma Xr{OE& àíZnÌ ewê$ H$aZo go nyd©

àýnÌ H$moS> d àýnÌ erf©H$ Om±M b|& Ho$bHw$boQ>a H$s AZw‘{V Zht h¡&

Section - A 8

×

2 = 16

(Very Short Answer Type Questions) (Compulsory)

Note: Answer all questions. As per the nature of the question you delimit your answer in one word, one sentence or maximum upto 30 words. Each question carries 2 marks.

IÊS> - "A'

(A{V bKw CÎmar¶ àíZ)

783

{ZX}e : g^r àíZm| Ho$ CÎma Xr{OE& Amn AnZo CÎma H$mo àíZmZwgma EH$ eãX, EH$ dm³¶ ¶m A{YH$V‘

³⁰

eãXm| ‘| n[agr{‘V H$s{O¶o& à˶oH$ àíZ

2

A§H$m| H$m h¡&

1) (i) An electric field in some region is given by E = axit+ byjt+ czkt, where a, b and c are constant. Find volume charge density.

{H$gr n[aga ‘o {dÚwV joÌ

^{E = axit+ byjt+ czkt}

go {X¶m OmVm h¡, Ohm±

^{a, b}

Ed§

^c

{Z¶Vm§H$ h¢& AmnVZ Amdoe KZËd kmV H$s{OE&

(ii) A charged partide of charge 1mc is moving with velocity 3jt in a region where electric field 2jt and magnetic field 4jt+ 5kt . Find the Lorents force acting on the charged particle. Here all units are in SI.

EH$ Amdo{eV H$U {OgH$m Amdoe ‘mBH«$mo Hy$bå~ h¡¡

^3jt

doJ go Eogo

^mJ ‘| J{V H$a ahm h¡ Ohm± {dÚwV joÌ

^2tj

VWm Mwå~H$s¶ joÌ

4tj+5kt

h¡ Vmo Amdo{eV H$U na bJZo dmbm bmaoÝO ~b kmV H$amo&

¶hm± g^r B©H$mB©¶m§

^SI

h¢&

(iii) Find the magnetic field in free space associated with the electric field vector E= E0Sin kz( - wt j)

‘w³V$ AmH$me ‘| {dÚwV joÌ

^{E= E0Sin kz( -wt j)}

Ûmam {X¶m OmVm h¡ Vmo BgHo$ gmW g§~Õ Mwå~H$s¶ j¡Ì H$m ‘mZ ³¶m hmoJm&

(iv) Define the Poynting Vector and write its formula.

nmo¶pÝQ>¨J g{Xe H$mo n[a^m{fV H$amo VWm BgH$m gyÌ {bImo&

(v) Express the field vectors E and B interms of the electro magnetic potential A and z.

joÌ g{Xe

^E

VWm

^B

H$mo {d^d ({dÚwV Mwå~H$s¶)

^A

VWm

^z

Ho$

ê$n ‘| 춺$ H$s{OE&

(vi) Explain the term "retar dedpotentials".

""‘pÝXV {d^dmo'' nX H$s ì¶m»¶m H$amo&

(vii) What is gauge invariance?

JoO {ZíMaVm ³¶m h¡?

(viii) Write down in variants of electro magnetic fields.

{dÚwV Mwå~H$s¶ joÌm| Ho$ {ZíMa {b{I¶o&

Section - B 4

×

8 = 32

(Short Answer Type Questions)

Note: Answer any four questions. Each answer should not exceed 200 words. Each question carries 08 marks.

(IÊS> - ~) (bKw CÎmar¶ àíZ)

{ZX}e : {H$Ýht Mma àíZm| Ho$ CÎma Xr{O¶o& Amn AnZo CÎma H$mo A{YH$V‘

200

eãXm| ‘| n[agr{‘V H$s{O¶o& à˶oH$ àíZ

⁰⁸

A§H$m| H$m h¡&

2) A conducting sphere of radius 'R' is placed in a uniform electric field 'E'₀ at the origin. By the method of images show that the potential at point ( , )r i is given by ( , ) E R cos

r r

0 r2

2

z i = _c - _m i

EH$

^'R'

{ÌÁ¶m H$m MmbH$ Jmobm EH$ g‘mZ {dÚwV joÌ

^'E'0

‘| ‘yb {~ÝXþna aIm hþAm h¡ à{V{~å~ {d{Y go àX{e©V H$s{OE H$s {~ÝXþ

^{( , )r i}

na {d^d H$m ‘mZ {ZåZ hmoJm

^{z( , )r i = E0cr-}_r^R22mcosi

3) Define relative permittivity er, electrical susceptibility Xe and write relation between them.

Amno{jH$ {dÚwVerbVm

^er

{dÚwVrd àd¥Îmr

^Xe

H$mo n[a^m{fV H$s{OE d

BZHo$ ‘ܶ gå~ÝY [b{I¶o&

4) Show that when a uniformly magnetised sphere in put in an external magnetic field Ho, then the magnetic field in side the sphere (Hin) is given by

Hin = Ho – M/3

where M is the magnetic moment per unit volume.

àX{e©V H$s{OE {H$ O~ EH$ g‘mZ ê$n go Mwå~H$sV JmoboH$mo, EH$ ~mø Mwå~H$Z joÌ

^Ho

‘| aIm OmVm h¡¡ Vmo Jmobo Ho$ AÝXa Mwå~H$s¶ joÌ

^(Hin)

{ZåZ go {X¶m OmEJm :

Hin = Ho – M/3

Ohm±

^M

EH$mH$ Am¶VZ H$m Mwå~H$s¶ AmKyU© h¡&

5) What is wave guide. Differentiate between Mettallic and dielectric wave guides. What are TE and TM waves?

Va§J nWH$ ³¶m h¡? KmpËdH$ Ed§ nam{dÚwVr¶ Va§J nW H$mo {d^o{XV H$s{OE&

^TE

Ed§

^TM

Va§J| ³¶m h¢?

6) What is continuity equation? Show that equation of continuity is contained in Maxwell's Equations.

gmV˶ g‘rH$aU ³¶m h¡? àX{e©V H$s{OE H$s gmV˶ g‘rH$aU ‘¡³gd¡b g‘rH$aUm| ‘| {Z{hV (em{‘b) h¡&

7) What is coulomb gauge? In a source free region if A= x i⁴t+ z t k^{2 2}t compute field vectors E and B and trans verse current JT .

Hy$bå~ Jm°O ³¶m h¡? ómoV ‘w³V joÌ ‘| ¶{X

^{A= x i4t+z t k2 2t}

hmo Vmo joÌ g{Xe

^{E, B}

Ed§ AZwàñWYmam

^JT

H$s JUZm H$amo?

8) Derive expression for Lienard-Wiechert potentials for a moving point charge. What is the significance of those potentials?

EH$ J{Verb {~ÝXþ Amdoe Ho$ {bE {bZmS>©-{dMQ>© [d^dm| H$m 춧OH$ ì¶wËnÞ H$amo& BZ {d^dm| H$m ³¶m ‘hËd h¡?

9) Derive relativistic equation of motion.

Amno{jH$s¶ J{VH$s g‘rH$aU ì¶wËnÞ H$amo&

Section - C 2

×

16 = 32 (Long Answer Type Questions)

Note: Answer any two questions. You have to delimit your each answer maximum 500 words. Each question carries 16 marks.

(IÊS> - g) (XrK© CÎmar¶ àíZ)

{ZX}e : {H$Ýht Xmo àíZm| Ho$ CÎma Xr{O¶o& Amn AnZo CÎma H$mo A{YH$V‘

500

eãXm| ‘| n[agr{‘V H$s{O¶o& à˶oH$ àíZ

¹⁶

A§H$m| H$m h¡&

10) (i) Show that the potential at any external point due to a charge distribution is given by the sum of the individual potentials due to monopoles dipoles, quadrupoles etc.

àX{e©V H$s{OE {H$ Amdoe {dVaU Ho$ Ûmam {H$gr ~mø {~ÝXþ na {d^d H$m ‘mZ EH$b Y«wd, {ÛY«wd MVwY«w©d Am{X Ho$ Ûmam CËnÞ {d^dm| Ho$ ¶moJ Ho$ Vwë¶ hmoVm h¡&

(ii) Show that the interaction energy W due to a charge distribution (Total charge q) in an eternal field E can be expressed as :

( ) ( ) (0) ...

W q o o iJ

x P E E

6 1

i J J

i

$ 2

z i 2

= - -

/ /

+

Where z, p and i are external potential, dipole moment vector and Quadrupole moment tensor respectively.

àX{e©V H$s{OE {H$ Amdoe {dVaU (Hw$b Amdoe

^q

) Ho$ Ûmam, ~mø {dÚwV joÌ

^E

‘|, Aݶmoݶ D$Om©

^W

H$mo {ZåZ n«H$ma go 춳V {H$¶m Om gH$Vm h¡:

( ) ( ) (0) ...

W q o o iJ

x P E E

6 1

i J J

i

$ 2

z i 2

= - -

/ /

+

Ohm±

^z,p

Ed§

ⁱ

H«$‘e… ~mø {d^d, {ÛY«wd AmKyU© g{Xe Ed§ MVwY«w©d

AmKyU© à{Xe h¢&

11) Discuss the propagation electro magnetic waves in a homogenous conducting medium and find out the expression for skin depth for it.

Give physical reasons for the rapid damping of the wave in such a medium.

{H$gr g‘m§Jr MmbH$ ‘mܶ‘ ‘| {dÚwV Mwå~H$s¶ Va§Jm| Ho$ g§MaU H$s {ddoMZm H$a| Ed§ BZHo$ {bE pñH$Z JhamB© Ho$ {bE 춧OH$ àmá H$s{OE& Eogo

‘mܶ‘ ‘| Va§Jm| Ho$ Vrd« Ad‘ÝXZ Ho$ ^m¡{VH$ H$maU ~VmB¶o&

12) (i) Starting from the expression for Lienard–Wiechart potentials for a point charge, obtain expression for the electric and magnetic fields due to an arbitrarity accelerated point charge.

{bZmS>©-{dMQ>© {d^dm| (EH$ {~ÝXþ Amdoe Ho$ {bE) go n«maå^ H$aVo hþE, EH$ ñdopÀN>H$ Ëd[aÌ {~ÝXþ Amdoe Ho$ Ûmam CËnÞ {dÚwV Ed§

Mwå~H$s¶ joÌm| Ho$ {bE 춧OH$ àmá H$s{OE&

(ii) Derive Larmor's formula for total power radiated by an accelerated charge.

EH$ Ëd[aV Amdoe Ho$ Ûmam CËg{O©V Hw$be{º$ Ho$ {bE bma‘a H$m gyÌ ì¶wËnÞ H$amo&

13) (i) Define electro magnetic field tensor and using electro magnetic field tensor derive the Lorentz transformation for electro magnetic field components.

{dÚwV Mwå~H$s¶ joÌ à{Xe H$mo n[a^m{fV H$a| VWm {dÚwV Mwå~H$s¶

joÌ à{Xe H$mo Cn¶moJ ‘| boVo hþE {dÚwV Ed§ Mwå~H$s¶ joÌm| Ho$

bmaoÊQ>O² ê$nmÝVaU ì¶wËnÞ H$amo&

(ii) Express Maxwell's equations in Covariant tensorform.

‘¡³gdob g‘rH$aUm| H$mo ghMa MVw{d©‘ Q>oÝga Ho$ ê$n ‘| 춺$ H$s{O¶o&