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1 Mathematical Physics JEST-6 Q. Give the coditio φ, the solutio of the equatio ψ φ φ is give by k. kφ kφ lφ kφ lφ ψ (b) ψ kφ (c) ψ (d) ψ As: Solutio: φ. φ. φ φ α + βy+ γz φ α + βy+ γz ( ) k φ. φ k α + β + γ kφ k ( ) If ψ ( α + βy+ γz) ψ ψ ψ y z ψ k α β γ ( ) ˆ ˆ ˆ ( ) ψ φ φ k. Q. The mea value of radom variable with probability desity As. : ( ) p ( + μ) ( ).ep σ π σ (b) μ is: (c) μ (d) σ Solutio: μ ep d ep d σ π σ σ fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

2 Q. Give a matri M π M, which of the followig represets cos 6 (b) 4 (c) 4 (d) As. : (b) Solutio: We have λ λ λ 4λ+ λ or λ For λ y gives Thus + y y. Takig, the eigevector associated with λ is For λ y y Takig, the eigevectors associated with λ is Thus π i M 6 / / M / / π i 6 / / π / / i 6 fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

3 e π i M 6 π i 6 e / / π i / / 6 e π π π π π π i i i i 6 6 i i e + e e + e 6 6 e e π π π π π π i i i i i i 6 6 e e e e e e + + i i πm πm + + cos i si i i i i πm πm cos i si i i Thus π M cos 6 4 Q4. The sum of the ifiite series is 5 7 π (b) π (c) π (d) 4 π As. : (d) Solutio: The series for ta for >, is give by π ta Puttig, we obtai π ta + + π π fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

4 Q5. A semicircular piece of paper is folded to make a coe with the cetre of the semicircle as the ape. The half-agle of the resultig coe would be: o 9 o (b) 6 o (c) 45 o (d) As. : (d) Solutio: Whe the semicircular piece of paper is folded to make a coe, the circumferece of base is equal to the circumferece of the origial semicircle. Let r be the radius of the base of the core ad be the radius of the semicircle. Hece, πr π r. The stay height of the come will also be. / Hece, siα Thus, α α r JEST-5 Q6. Give a aalytic fuctio f ( z) φ(. y) + iψ (, y), where (, y) + 4 y + y As. : (c) If C is a costat, which of the followig relatios is true? ψ (, y) y+ 4y+ C (b) ( ) ψ, y y + C (c) ψ (. y) y+ 4y + C (d) ( ) Solutio: ( ) u Q, y + 4 y + y ψ, y y + C φ. From C.. equatio u v y u y v u + 4 fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 4

5 v + 4 y ( ) v y+ 4y+ f (i) u y + y v + y ( ) v y+ + f y ( ) ( ) y+ 4y+ f y + f y (ii) ( ) ( ) f, f y 4y v y+ 4y + c Q7. If two ideal dice are rolled oce, what is the probability of gettig at least oe 6? As: 6 (b) 6 Solutio: Number of poit i sample space ( S ) (c) 6 (,6 ),(,6 ),(,6 ),( 4,6 ),( 5,6 ),( 6, ),( 6, ),( 6, ),( 6, 4 ),( 6,5 ),( 6,6) Number of poit i populatio ( P ) 6 6 ( ) ( ) S Probability that at least oe si o face of dice P 6 (d) 6 5 fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 5

6 Q8. What is the maimum umber of etrema of the fuctio f ( ) P ( ) (, ) ad ( ) P k is a arbitrary polyomial of degree k? k e where As: k + (b) k + 6 (c) k + (d) k (c) Solutio: ( ) ( ) f P e ( ) ( ) + ( ) ( + ) f P P e 4 + f ( ) ( )( ) ( ) P P + is polyomial if order k + From the sig scheme maimum umber of etrema k + s e Q9. The Beroulli polyomials B () s are defied by, B () s e! followig relatios is true? e e ( s ) B +! () s ( ) (b) ( ). Which oe of the s e B e +! ( s)( ) ( ) (c) e e ( s ) B ( s)( )! (d) e e ( s ) ()( s ) B! As: (d) S e e Solutio: B ( S) Put S ( S ), ( ) ( ) ( ) ( ) B S B S S e B e ( S ) ( ) S e B e ( S)( ) fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 6

7 Q. Cosider the differetial equatio G ( ) + kg( ) δ ( ) followig statemets is true? Both G ( ) ad ( ) (b) G ( ) is cotiuous at but G ( ) is ot. (c) G ( ) is discotiuous at. (d) The cotiuity properties of G( ) ad ( ) G are cotiuous at. As: (c) Q. 99 The sum is equal to m m + + m 9 (b) As: Solutio: 99 m m+ + m m+ m m+ m m+ m m ( ) ; where k is a costat. Which G at depeds o the value of k. 99 (c) ( 99 ) m JEST-4 f f + f? Q. What are the solutios to ( ) ( ) ( ) (d) c e / (b) c + c / (c) ce + c (d) As.: (d) Solutio: Auilary equatio D + D D +, + D ( ) oots are equal the f ( ) ( c+ c) e ( ) + f ce c e c e + c e Q. The value of. fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / e d by usig the oe-segmet trapezoidal rule is close to.67 (b).87 (c).99 (d) 4.9 As.: (c) h Solutio: h.. I y(.) + y(.).99 y e 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 7

8 Q4. Give the fudametal costats (Plack s costat), G (uiversal gravitatio costat) ad c (speed of light), which of the followig has dimesio of legth? G c G (b) 5 c G (c) c (d) c 8πG As.: [ ][ ] Solutio: ML T M L T [ ] L L L T gr ML T, G [ M L T ] m Q5. The Laplace trasformatio of e t si 4t is s 4 + 4s + 5 (b) s 4 + 4s + (c) s 4s + 4s + (d) s 4s + 4s + As.: (b) Solutio: L e at si bt b ( ) s+ a + b t L e si 4t ( ) 4 s s 4 + 4s+ Q6. Let us write dow the Lagragia of a system as L( ) m k c dimesio of c? fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / ,, + +. What is the MLT (b) MT (c) MT (d) ML T As.: (c) Solutio: Accordig to dimesio rule same dimesio will be added or subtracted the dimesio of M dimesio of C [ ] [ ] ML T C L LT [ ] [ ML T ] [ L M ] C [ MT ] 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 8

9 Q7. The Dirac delta fuctio δ ( ) satisfies the relatio ( ) ( ) d f ( ) As.: (d) behaved fuctio f ( ). If has the dimesio of mometum the δ ( ) has the dimesio of mometum (b) δ ( ) has the dimesio of ( mometum ) (c) δ ( ) is dimesioless (d) δ ( ) has the dimesio of ( mometum) Solutio: f ( ) δ ( ) d f ( ) f ( ) δ ( ) d f ( ) [ f ( ) ] δ ( ) P f ( ) Sice, [ f ( ) ] [ f ( ) ] If ( ) α + β ( ) β Q8. The value of limit [ ] δ ( ) [ ] P F is force [ M LT ] F is also [ M LT ] z + lim z i z 6 + is equal to (b) (c) -/ (d) 5/ As.: (d) f δ for a well Solutio: z + z lim i z 6 lim + z i 6z 9 z lim 5 z 5 z i Q9. The value of itegral si z I dz c z π with c a circle z, is (b) π i (c) π i (d) π i fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 9

10 As.: (c) si z π Solutio: I pole z π z C z π π esidue at z z so it will be lies withi the cotour iz e I( ) emg πi C π z es Now π z π iz z e iπ / e i π z I π i π i (takig imagiary part) ; esidue JEST- Q. A bo cotais cois out of which 99 are fair cois ad is a double-headed coi. Suppose you choose a coi at radom ad toss it times. It turs out that the results of all tosses are heads. What is the probability that the coi you have draw is the doubleheaded oe?.99 (b).95 (c).75 (d). As.: (c) e( z ) + Im( z ) Q. Compute lim z z The limit does ot eist. (b) (c) i (d) - As.: Solutio: ( ) ( ) e z + Im z y + y y + y lim lim lim z y + iy y + iy z z y fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

11 y + y lim y y + iy ad + y y lim y y + iy i Q. The vector field ziˆ + yˆj i cylidrical polar coordiates is ˆ ρ( z cos φ + si φ) e ρ + ρ siφ cosφ( z) e φ ˆ (b) ρ( z cos φ + si φ) e ρ + ρ siφ cosφ( + z) e φ ˆ (c) ρ( z si φ + cos φ) e ρ + ρ siφ cosφ( + z) e φ ˆ (d) ρ( z si φ + cos φ) e ρ + ρ siφ cosφ( z) e φ As.: Solutio: A ziˆ + yˆj A z, A y, A y z A A eˆ A ˆ eˆ + A yˆ eˆ + A z eˆ ( ) ( ) ( ˆ ) ρ ρ ρ y ρ z ρ Aρ ρcosφz( cosφ) + ρsiφ( siφ) + ρ ( ρcosφ + ρsi φ) A Ae ˆ A e ˆ ˆ + A ye ˆ ˆ + A ze ˆ ( ) ( ) ( ˆ ) φ φ φ y φ z φ ˆ ˆ ˆ ˆ A z e ˆ ρ A cos ( si ) z+ si cos A ρcosφ siφ( z) eˆ φ ρ φ φ ρ φ φ φ A Aeˆ + Aeˆ + Aeˆ ρ φz+ φ eˆ + ρ φ φ z eˆ ( cos si ) cos si ( ) ρ ρ φ φ z z ρ φ Q. There are o average buses per hour at a poit, but at radom times. The probability that there are o buses i five miutes is closest to.7 (b).6 (c).6 (d).9 As.: (d) Q4. Two druks start out together at the origi, each havig equal probability of makig a step simultaeously to the left or right alog the ais. The probability that they meet after steps is!! (b) (c)! (d)! 4!! 4 As.: φ fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

12 Solutio: Ito probability of takig ' r ' steps out of N steps r N r N C r total steps N + for takig probability of steps out of N N N!!! P NC ( N ) ( )!!!!! 4 Q5. What is the value of the followig series? As.: (d) ! 4!! 5! (b) e (c) e (d) Solutio: 4 5 θ θ θ θ cosθ +..., si θ θ +...! 4!! 5! cos + si si θ + cos θ! 4!! 5! / λ Q6. If the distributio fuctio of is f ( ) e over the iterval < <, the mea value of is λ (b) λ (c) λ (d) As.: (b) Solutio: it is distributio fuctio so ( ) λ f d. e d f ( ) d λ e d λ e λ e d λ d fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

13 Q7. The value of the itegral As. : (b) (b) l + ( ) π 4 d JEST- is (c) π (d) π Solutio: l l z d ( + ) ( z + ) dz Let us cosider ew fuctio f ( z) Pole at z esidue at z l z z ± i is simple pole of secod order. iis ( l z) ( ) ( + ) d ( z i ) dz z i z i ( l z) ( + ) d dz z i +, the I l z z + dz A B r z ( z+ i) l ( z). ( l z).( z+ i) ( z+ i) 4 l l. z ( z+ i) ( z) ( z) ( z+ i) l l i ( i) i ( i) ( i) π π es + i z i 4 6 iπ iπ 4 8i π π i + 8i Similarly at z i; es i z i fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / π π 4 6 l z π π π π I dz π i i i π i + z π i f z dz f z dz ( ) ( ) ; vaish ABr AB 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com AB

14 Alog path A; z + iε ad alog path B; z iε Thus π ( ) l ( ε) ( ε) l ( ε) ( ε) + i i i f z dz d d AB + i + i + i i π + i + i + l ( + ε) ( ε) l ( ε) ( ε) i d d ( ) π l ( ) l + i iπ π i d d ε + + ; ( l ( ) + iπ) ( l ( ) iπ) l 4π ( + ) ( + ) π i d i l iπ π 4π i 4 ( + ) Q8. If [] deotes the greatest iteger ot eceedig, the [] e e As.: (b) (c) e e Solutio: [ ] < [], < [], < [ ] ow [] e d [] e d + [] e d + [] e d + [] e d + e. e d +. e d + 4. e d 4 [ e ] + ( e ) + ( e )... e e e + e e + 4e 4e d (d) e e e + e + e + e fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 4

15 e e fiziks e e r e e e + Q As, the ifiite series diverges (b) coverges to uity (c) coverges to π / 4 (d) oe of the above As.: (c) 5 7 π Solutio: ta ta Q. What is the value of the followig series? ! 4!! 5! (b) e (c) e (d) As.: (d) Solutio: e ,!! e +!...! e cosh + e +! + 4! +... sih ( e e ) ! 5! i.e cos h si h Q. A ubiased die is cast twice. The probability that the positive differece (bigger - smaller) betwee the two umbers is is / 9 (b) / 9 (c) / 6 (d) / As.: p Solutio: ( ) ( E) ( S ) fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 5

16 The umber of ways to come positive differece [(,), ( 4, ),( 5, ), ( 6, 4) ] p 4 6 ( ) 9 Q. For a N N matri cosistig of all oes, all eigevalues (b) all eigevalues (c) the eigevalues are,,., N (d) oe eigevalue N, the others As.: (d) Solutio:, so far,, N N matri oe eige value is N ad aother s eige value is zero fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 6

17 Classical Mechaics JEST-6 Q. A hoop of radius a rotates with costat agular velocity ω about the vertical ais as show i the figure. A bead of mass m ca slide o the hoop without frictio. If g < ω a at what agle θ apart from dθ d θ ad π is the bead statioary (i.e., )? dt dt π g g taθ (b) siθ ω a ω a g g (c) cosθ (d) taθ ω a πω a As: (c) Solutio: ( si ) L ma θ + θφ + mgacosθ d L L ma θ ma siθcosθφ + mga siθ dt θ θ ( ) ( ) dθ d θ ma siθcosθφ + mga siθ dt dt, φ ω ad g g < ω athe cosθ ω a a θ fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

18 Q. The cetral force which results i the orbit r a( cosθ ) to: + for a particle is proportioal r (b) r As: (c) Solutio: r a( + cosθ ) u r a ( + cosθ ) (c) r If J is agular mometum ad m is mass of particle (d) Noe o the above J d u du siθ + u f m dθ u dθ a + ( cosθ ) du si θ cosθ + dθ a θ a θ ( + cos ) ( + cos ) J d u J si θ cosθ + u + + f m dθ m a( + cosθ) a( + cosθ) a( + cosθ ) u J cos θ cos θ + + f m a( cosθ) a( cosθ) a( cosθ ) + u + + Put u a f u u ( + cosθ ) so ( ) f r,cosθ au r ad solvig fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

19 Q. Light takes approimately 8 miutes to travel from the Su to the Earth. Suppose i the frame of the Su a evet occurs at t at the Su ad aother evet occurs o Earth at t miute. The velocity of the iertial frame i which both these evets are simultaeous is: 8 c with the velocity vector poitig from Earth to Su (b) 8 c with the velocity vector poitig from Su to Earth : - (c) The evets ca ever be simultaeous - o such frame eists (d) c 8 with velocity vector Poitig from to Earth As: Solutio: ' ' c 8 6 ' ' t 6 t ' ' ' v ' v t + t + c c ' ' v ' ' ( ) t t t t + v v c c c ( ) ' ' ' ' t t+ v 6 + v c 8 6 v c c c 8 c Negative sig idicate frame is movig with the velocity vector poitig from Earth to 8 Su. fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

20 Q4. For the coupled system show i the figure, the ormal coordiates are + ad correspodig to the ormal frequecies ω ad ω respectively. k k k m m As: (b) At t, the displacemets are A,, ad the velocities are v v. The displacemet of the secod particle at time t is give by: ( ) A A () t cos( ωt) + cos( ωt) (b) () t cos( ωt) cos( ωt) ( ) ( ) A A (c) () t si ( ωt) si ( ωt) (d) () t si ( ωt) si ( ωt) Solutio: Usig boudary coditio at t ad v ( ) A Oly () t cos( ωt) cos( ωt) will satisfied Q5. A cylidrical shell of mass i has a outer radius b ad a ier radius a. The momet of iertia of the shell about the ais of the cylider is: mb a mb a ( ) (b) ( + ) (c) mb ( + a ) (d) mb ( a ) As: (b) ( b a ) ( ) b m m dm π d b + a a π a b fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 4

21 JEST-5 Q6. The distace of a star from the Earth is 4. 5 light years, as measured from the Earth. A space ship travels from Earth to the star at a costat velocity i 4. 5 years, accordig to the clock o the ship. The speed of the space ship i uits of the speed of light is, (b) (c) (d) As: (b) Solutio: Proper life-time Δ t, Δ t c v Δt Δ t v c v / c Q7. A classical particle with total eergy E moves uder the ifluece of a potetial As: (, y) + y + y y V +. The average potetial eergy, calculated over a log time is equal to, E E (b) Noe of the above is correct. Solutio: If oe will use virial theorem the T (c) 5 E (d) E 5 V if V r accordig to problem So E T + V E V + V V E 5 But virial theorem is used oly for coservative forces. Force coservative F where F V V (, y) + y+ y + y V ( 9 + y+ y ) iˆ+ ( + 4y+ y ) ˆj F i.e. o coservative i ature. So we caot use viral theorem. Therefore, oe of the aswer is correct fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 5

22 Q8, A chai of mass M ad legth L is suspeded vertically with its lower ed touchig a weighig scale. The chai is released ad falls freely oto the scale. Neglectig the size of the idividual liks, what is the readig of the scale whe a legth of the chai has falle? Mg Mg Mg 4Mg (b) (c) (d) L L L L As: (c) dp Mg d( Δmv) Mg Solutio: eadig of scale impulse + actual weight + + dt L dt L M d Mg Mv Mg Mg Mg Mg v L dt L L L L L L v g ad Q9. A bike stutma rides iside a well of frictioless surface give by z a( + y ) As: the actio of gravity actig i the egative z directio. g gzˆ maitai to be able to ride at a costat height z without fallig dow? gz (b) gz (c) gz M Δ m d L, uder What speed should he (d) The biker will ot be able to maitai a costat height, irrespective of speed. (c) Solutio: z a( + y ) Usig equatio of costrai, we must solve the give system i cylidrical co-ordiate. z arr ar z ( ) L m r + r θ+ z mgz ( ) ( 4 ) ( 4 ) L m r + r θ+ a r r mgar m r + a r + r θ fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 6

23 Equatio of motio d L L dt r r ( ) mr ra r mr arr mr mgar θ + z z, r, r r At So θ mr θ mgar ga θ ga, / z ga v gz a fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / v ga r, v ga r ( z ar ) Q. The Lagragia of a particle is give by L q qq. Which of the followig statemets As: Solutio: is true? This is a free particle (b) The particle is eperiecig velocity depedet dampig (c) The particle is eecutig simple harmoic motio (d) The particle is uder costat acceleratio. L q qq d L L dt q q L d L q q q q q dt q dq dq q q + q q C q Ct+ α dt dt Q. How is your weight affected if the Earth suddely doubles i radius, mass remaiig the same? Icreases by a factor of 4 (b) Icreases by a factor of (c) Decreases by a factor of 4 (d) Decreases by a factor of As: 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 7

24 GM Solutio: W m ad W m GM ( ) W Q. A sprig of force costat k is stretched by. It takes twice as much work to stretch a secod sprig by. The force costat of the secod sprig is, k (b) k (c) 4 k (d) 8 k As: (d) Solutio: The relatio betwee eergy ad maimum displacemet is E ka For A E ; E E k ad For A ; W 4 E k k 8 k k k 8k k 8k 8 JEST-4 Q. A dyamical system with two geeralized coordiates q ad q has Lagragia L + q q. If is give by p ad p are the correspodig geeralized mometa, the Hamiltoia ( p + )/ 4 (b) ( q + )/ 4 (c) ( )/ p As.: Solutio: H q ipi L L q H p p q q p + q p q q p p L p ad p 4 p p + 4 L q p H p + (d) ( p q p )/ 4 p q q ( p + p ) 4 p + q fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 8

25 Q4. I a certai iertial frame two light pulses are emitted, a distace 5 km apart ad separated by5 μs. A observer who is travelig, parallel to the lie joiig the poits where the pulses are emitted, at a velocity V with respect to this frame otes that the pulses are simultaeous. Therefore V is As.: (c).7c (b).8c (c).c (d).9c Solutio: 5 m, t t 5 sec 6 t v v v t + t + ( t t) ( ) c c c t v v v c c c t t 6 v 5 5 v.c c Q5. A double pedulum cosists of two equal masses m suspeded by two strigs of legth l. What is the Lagragia of this system for oscillatios i a plae? Assume the agles θ, θ made by the two strigs are small (you ca use cos θ θ / ). Note: ω g / l. L ml (b) L ml (c) L ml (d) L ml θ + θ ωθ ωθ θ + θ + θ θ ωθ θ + θ θ θ ωθ θ + θ + θ θ ωθ ωθ ωθ ω θ fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 9

26 As.: (b) Solutio: l siθ, y l cosθ + lsiθ y y + lcosθ θ l lsiθ + lsi θ, y lcosθ + lcosθ lcosθ θ + lcos θ θ, y lsiθθ lsiθ θ θ l θ θ θ θ θ θ θ θ y l cos + l cos + l cos cos l si l siθ θ + l siθ siθ θ θ θ θ m ( ) + y l θ + l θ + l cos θ θ θθ also + y l θ L T V m( + y + + y ) mgy mgy L m( l θ + l θ + l θ + l cos( θ θ) θθ ) + mglcosθ + mglcosθ g θ g θ L ml θ + θ + θθ + + l l ( θ θ ) g g θ g g θ L ml θ + θ + θθ + + l l l l comparig give optios, optio (b) is correct i.e. ωθ L ml θ + θ + θθ ω θ 4 Q6. A moochromatic wave propagates i a directio makig a agle cos o 6 with the -ais 4c i the referece frame of source. The source moves at speed v towards the 5 observer. The directio of the (cosie of agle) wave as see by the observer is cos θ (b) 4 As.: cos θ (c) 4 cos θ (d) 6 cos θ Solutio: 4c v, 5 u c o ccos 6, u y csi 6 c o fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

27 Now u fiziks c 4 + c 5 c 4c + 5 c c 4 cosθ 4 Q7. The acceleratio eperieced by the bob of a simple pedulum is maimum at the etreme positios (b) maimum at the lowest (cetral) positios (c) maimum at a poit betwee the above two positios (d) same at all positios As.: Solutio: T si θ ma, T cos θ mg o a g taθ at θ 9 a is maimum at etreme positio.. 4 Q8. Cosider a Hamiltoia system with a potetial eergy fuctio give by ( ) Which of the followig is correct? The system has oe stable poit (c) The system has three stable poits As.: V 4 Solutio: ( ) V 4 [ ] ±, V V d 4 d < ± θ l T T cosθ T siθ mg (b) The system has two stable poits (d) The system has four stable poits V V For stable poit V ad > V > fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

28 Q9. Two poit objects A ad B have masses Kg ad Kg respectively. They are iitially at rest with a separatio equal to m. Their mutual gravitatioal attractio the draws them together. How far from A s origial positio will they collide? / m (b) / m (c) / m (d) /4 m As.: (d) Solutio: Sice gravitatioal force is coservative, therefore they collide at their cetre of mass m ( ) m m m A B ( ) m 4 JEST- Q. I a observer s rest frame, a particle is movig towards the observer with a eergy E ad mometum P. If c deotes the velocity of light i vacuum, the eergy of the particle i aother frame movig i the same directio as particle with a costat velocity v is As.: ( E + vp) ( v / c) (b) ( E vp) ( v / c) (c) ( E + vp) ( v / ) [ c ] (d) ( E vp) ( v / ) [ c ] v v v t+ Solutio: c + + t c c c ct, ct c v v v c c c E E+ v Now, c E E+ Pv E E E E mc, E Pc P E c v v c c Q. The free fall time of a test mass o a object of mass M from a height to is As.: π (b) GM ( / + ) (c) ( / ) GM π (d) π GM GM fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

29 Solutio: Equatio of motio md dt r GMm d r GM d r A GM r dt r dt r A dv A dr d v d A v A v + C dt r dt dt dt r r whe r, v A A v A A A A dr A r + C C v r r dt r r dr r A t dt put r u, dr udu whe r, r, u, u u u udu A t dt A t u u du A t u u u + si A t + si + si A π π π π t t + t + A GM + 4 A GM Q. Uder a Galilea trasformatio, the coordiates ad mometa of ay particle or system trasform as: t ' t, r ' r + vt ad p ' p + mv where v is the velocity of the boosted frame with respect to the origial frame. A uitary operator carryig out these trasformatios for a system havig total mass M, total mometum P ad cetre of mass coordiate X is (c) As.: (b) i M v. X / i t v. P / e e (b) i M v. X / i t v. P / i M v t /( ) e e e (d) e e i M v. X / i t v. P / e e i t v. P / i M v t / e ( ) i M v t / ( ) fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

30 Q. A spherical plaet of radius has a uiform desity ρ ad does ot rotate. If the plaet is made up of some liquid, the pressure at poit r from the ceter is 4πρ G ( r ) πρ G (c) ( r ) 4πρG (b) ( r ) ρg (d) ( r ) As.: (c) r ρ 4πr drgm dm g dm g Solutio: Pressure dp dp dp A 4πr 4πr dr r dm( mass of elemetary part) dp 4π ρ 4πr drg ρ 4πr r 4π dp ρ Grdr 4π 4π r 4π r dp ρ Grdr p ρ G p ρ G r r 4π ρ G π p ( r ) p ρ G( r ) Q4. A particle of mass m is throw upward with velocity v ad there is retardig air resistace proportioal to the square of the velocity with proportioality costat k. If the particle attais a maimum height after time t, ad g is the gravitatioal acceleratio, what is the velocity? k g g ta t gk ta g (b) t k k As.: (c) (c) g ta ( gkt) (d) gk ta ( gkt ) k fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 4

31 Solutio: Equatio of motio g dv k + v m dt fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / mdv dv k dv + + dt dt m g+ v m k m mg kv g v dt k dv gm + v k dt Q5. Cosider a uiform distributio of particles with volume desity i a bo. The particles As.: (c) m k gm k have a isotropic velocity distributio with costat magitude v. The rate at which the gm k particles will be emitted from a hole of area A o oe side of this bo is va (b) A v (c) t 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 5 ta v A v (d) oe of the above 4 Q6. If, i a Kepler potetial, the pericetre distace of particle i a parabolic orbit is r p while As.: the radius of the circular orbit with the same agular mometum is r c, the r c r p (b) r c r p (c) r c r p (d) rc rp Solutio: Ioic equatio l r p l r + ecosθ for parabola e for circle, e, θ l + e,, l r, l r r r r C p C p C Q7. A K meso (with a rest mass of 494 MeV) at rest decays ito a muo (with a rest mass of 6 MeV) ad a eutrio. The eergy of the eutrio, which ca be massless, is approimately MeV (b) 6 MeV (c) MeV (d) 88 MeV As.: (b) Solutio: k μ+ ν, c ( mk m ) c μ c c c c Eν m 494 k c MeV 988

32 Q8. A light beam is propagatig through a block of glass with ide of refractio. If the glass is movig at costat velocity v i the same directio as the beam, the velocity of the light i the glass block as measured by a observer i the laboratory is approimately c c u + v (b) u v c (c) u + v + (d) As.: c v + Solutio: ow u v c + c v + c + v c c u c v v v + + c c v v c v cv c v u + v c c c c Q9. The period of a simple pedulum iside a statioary lift is T. If the lift accelerates As.: (c) dowwards with a acceleratio 4 g, the period of the pedulum will be T (b) T / 4 (c) T / (d) T / 5 Solutio: T l π lift accelerates dow wards the g T l π T g g π l g g 4 4l π π g l g T T fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 6

33 Q. The velocity of a particle at which the kietic eergy is equal to its rest eergy is (i As.: Solutio: terms of c, the speed of light i vacuum) c / (b) c / 4 (c) / 5c (d) c / K. E mc m c, rest mass eergy K.E. rest mass eergy mc mc mc m c m c m c mc v v c c Q. If the Poisso bracket {, } As.: m c v 4 c v 4 v c p, the the Poisso bracket { p, p} c + is? (b) (c) (d) + {, p} + {, p} + ( ) ( ) Solutio: { p, p} {, p} + { p, p} Q. The coordiate trasformatio.8 +.6y, y.6. 8y + represets a traslatio (c) a reflectio As.: (b) (b) a proper rotatio (d) oe of the above fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 7

34 Q. A small mass M hags from a thi strig ad ca swig like a pedulum. It is attached above the widow of a car. Whe the car is at rest, the strig hags vertically. The agle As.: (b) made by the strig with the vertical whe the car has a costat acceleratio is approimately (b) 7 (c) Solutio: Tsiθ ma, T cos θ mg, taθ a g 5 (d) 9 a. θ ta ta g 9.8 a. m/ s JEST- Q4. For small agular displacemet (i.e., siθ θ), a simple pedulum oscillates harmoically. For larger displacemets, the motio becomes a periodic (b) remais periodic with the same period (c) remais periodic with a higher period (d) remais periodic with a lower period As. : (c) Q5. A plaet orbits a massive star i a highly elliptical orbit, i.e., the total orbital eergy E is close to zero. The iitial distace of closest approach is. Eergy is dissipated through tidal motios util the orbit is circularized with a fial radius of f. Assume that orbital agular mometum is coserved durig the circularizatio process. the As. : (d) f / (b) f (c) f (d) f Solutio: For elliptically motio E mr J + mr GMm r E ad closest approach is at r J + m GMm fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 8

35 J GMm M J GMm from coditio of circular orbit J m f f () r V r J m GMm f f GMm m f f GMm f Q6. A biary system cosists of two stars of equal mass m orbitig each other i a circular orbit uder the ifluece of gravitatioal forces. The period of the orbit is т. At t, the motio is stopped ad the stars are allowed to fall towards each other. After what time t, epressed i terms of т, do they collide? d α α α + si α τ (b) τ (c) τ (d) 4 τ As. : (d) d GMm Solutio: M dt d GM dt d A dt dv dt A v d dt fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 9

36 d dt fiziks v d dt A v A + C where v v v A A A d dt A d t A dt put u d udu,, u, u u du u t A dt u u + si u A t + A + si t A si t t A π fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

37 t π A t π () GM ad mv GM v π v τ 4π τ 4π GM GMm GM τ π π τ τ GM GM τ τ t 4 Q7. I a certai itertial frame two light pulses are emitted at poit 5 km apart ad separated i time by 5 μs. A observer movig at a speed V alog the lie joiig these poits otes that the pulses are simultaeous. Therefore V is.7c (b).8c (c).c (d).9c As. : (c) Solutio: Δt, t t 5μs, 5km v V t t t V + C V C V t + C V C fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

38 ( t t ) ( ) V C V C 5 6 V C 5 6 V V C. C C 5 Q8. A jet of gas cosists of molecules of mass m, speed v ad umber desity all movig co-liearly. This jet hits a wall at a agle θ to the ormal. The pressure eerted o the wall by the jet assumig elastic collisio will be cos p mv θ (b) p mv cosθ (c) ( / ) mv cos θ p (d) p mv As.: Solutio: chage i mometum alog chage i mometum alog Δ p mv cosθ y directio will be cacelled out directio Pressure Force Area Δp Δt A Δp A. Δt mv cosθ v cosθn Pressure p v p mv cos θ fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / Δp Δpv cosθ L A. L A v cosθ N, V,, ( V Area L A L) Q9. If the coordiate q ad the mometum p from a caoical pair (q, p), which oe of the sets give below also forms a caoical? (q - p) (b) (q, p ) (c) (p, - q) (d) (q, - p ) As.: (c) Solutio: for caoical pair ( p, q) p ( q) ( p) ( q) ( ) q p p q 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

39 Q4. A girl measures the period of a simple pedulum iside a statioary lift ad fids it to be T secods. If the lift accelerates upward with a acceleratio g / 4, the the pedulum will be As.: (c) T (b) T / 4 (c) T 5 (d) T 5 Solutio: T π l g Sice lift accelerated upward the T π l g + g T π l g + g 4 l T π 4 5g l T π g 5 T T 5 fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

40 Electromagetic Theory JEST-6 Q. The maimum relativistic kietic eergy of β particles from a radioactive ucleus is equal to the rest mass eergy of the particle. A magetic field is applied perpedicular to the beam of β particles, which beds it to a circle of radius. The field is give by: mc e As: (c) Solutio: mc (b) e KE mc m c m c m m ma m m m m v c v v c c mc (c) e mc (d) e mv eb mv m m c e e e B c Q. The stregth of magetic field at the ceter of a regular heago with sides of legth a carryig a steady curret I is: μi 6μI (b) π a π a As4: (d) d acos a B μ I 4π d ( siθ siθ ) μi (c) π a C a d 6 I μi μi μi B si si 4π d π a 4π a μi (d) π a I a C μ I μ I μ I B 6B 6 πa πa π a fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

41 Q. A spherical shell of radius carries a costat surface charge desity σ ad is rotatig about oe of its diameters with a agular velocityω. The magitude of the magetic momet of the shell is: 4 4 4πσω 4πσω (b) (c) As : (b) Solutio: The total charge o the shaded rig is dq σ (π si θ) dθ π Time for oe revolutio is dt ω 4πσω 5 dq Curret i the rig I σω si θdθ dt Area of the rig π( si θ ), so the magetic momet of the rig is dm ( σω si θdθ) π si θ 4 (d) 4πσω 9 4 z ω θ dθ siθ dθ 4 m σω θdθ π σω 4 π 4 si Q4. The electric field ( ω ) 4π 4 m σω zˆ π E E ˆ ˆ si t kz + Esi ωt kz+ y a liwearly polarized wave (b) a right-had circularly polarized wave (c) a left-had circularly polarized wave (d) a elliptically polarized wave As: (d) represets: fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

42 Q5. Suppose yz plae forms the boudary betwee two liear dielectric media I ad II with dielectric costat I ad II 4 the iterface is give by E 4ˆ+ yˆ+ 5zˆ I, respectively. If the electric field i regio I at, the the electric field E II regio II is: 4ˆ+ yˆ+ 5zˆ (b) 4ˆ+.75yˆ.5zˆ (c) ˆ+ yˆ+ 5zˆ (d) ˆ+ yˆ+ 5zˆ As: (d) at the iterface i Solutio: E E E yˆ + 5zˆ ad I II II E ˆ+ yˆ+ 5zˆ II E II εi εi E II E I 4 ˆ ˆ E ε ε 4 I II II Q6. How much force does light from a.8 W laser eert whe it is totally absorbed by a object? (b) 9 6. N (c) 9.6 N (d) 8.6 N N As: Solutio: adiatio Pressure F I P F P.8 9 F 6. N 8 A c Ac c Q7. Self iductace per uit legth of a log soleoid of radius with turs per uit legth is: μπ (b) μπ (c) As: μπ (d) μπ fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

43 JEST-5 Q8. A circular loop of radius, carries a uiform lie charge desity λ. The electric field, calculated at a distace z directly above the ceter of the loop, is maimum if z is equal to, (b) (c) (d) As: (b) Solutio: E 4πε ( λ π ) z ( ) / + z ( / ) For maimum E de λ π + z z / + z z, dz 4πε ( + z ) ( ) / + z z + z + z z z z Q9. Cosider two poit charges q ad λ q located at the poits, a ad μa, respectively. Assumig that the sum of the two charges is costat, what is the value of λ for which the magitude of the electrostatic force is maimum? μ (b) (c) (d) + μ μ As: (b) ( λq q) λq λc Solutio: F 4πε μa a 4πε a μ 4πε a μ + λ ( ) ( ) ( ) ( ) q+ λq c df ( + λ) c λc ( + λ) For maimum F, 4 dz 4πε ( ) ( ) a μ + λ ( + λ) c λc ( + λ) + λ λ λ fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 4

44 Q. A spherical shell of ier ad outer radii a ad b, respectively, is made of a dielectric k material with froze polarizatio P() r rˆ,where k is a costat ad r is the distace r from the its cetre. The electric field i the regio a < r < b is, k k E rˆ (b) E rˆ r r (c) E k (d) E rˆ r As: (b) k ˆ k k + P.r (at r b) b Solutio: ρ b.p r ad σ r r b P.ˆ r r k P.r ˆ (at r a) a k r k For a< r <b ; Qec 4π a + 4πr dr a a 4πka 4πk( r a) 4π kr r Q E ec 4πε r k E rˆ ε r λ r e Q. The electrostatic potetial due to a charge distributio is give by V () r A where r A ad λ are costats The total charge eclosed withi a sphere of radius λ, with its origi at r is give by, As: 8π A e 4π A (b) e π (c) A e (d) Solutio: V( r) λr e A r ( λ ) λr λr λr re e Ae E V A rˆ ( + λr) rˆ r r π π λr Ae Q ε Eda. ε + λr rr. siθdθdφr 4πε Ae + λr ec λr ( ) ˆ ˆ ( ) r fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 5

45 Thus total charge eclosed withi a sphere of radius Q ec r is λ λ 8 A 4 Ae λ πε πε + λ λ e Q. The ski depth of a metal is depedet o the coductivity ( σ ) of the metal ad the agular frequecy ω of the icidet field. For a metal of high coductivity, which of the followig relatios is correct? (Assume that permittivity of the medium.) σ >> ω, where is the electrical σ d (b) ω d σω (c) d σω (d) d ω σ As: (b) Solutio: Ski depth d σμω Q. The wavelegth of red helium-eo laser i air is68 A. What happes to its frequecy i glass that has a refractive ide of. 5? Icreases by a factor of (b) Decreases by a factor of. 5 (c) emais the same (d) Decreases by a factor of.5 As: (c) Solutio: Frequecy of electromagetic wave does ot chage whe it eter i medium of ay refractive ide. o fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 6

46 Q4. The approimate force eerted o a perfectly reflectig mirror by a icidet laser beam of power mw at ormal icidece is N (b) N (c) 9 N (d) 5 N As: (b) Solutio: Whe electromagetic wave is reflected by mirror the mometum trasferred to the mirror per uit area per secod is twice the mometum of the light strikig the mirror per uit area per secod dp Power i.e. 6.6 kg m / s 8 dt c dp The force eerted o the reflectig mirror is F 6.6 N dt Thus best suitable aswer is optio (b). Q5. Which of the followig epressios represets a electric field due to a time varyig magetic field? K( ˆ + yyˆ + zzˆ ) (b) K( ˆ + yyˆ zzˆ ) (c) K( ˆ yyˆ ) (d) K( yyˆ yˆ + zzˆ ) As: (d) Solutio: For time varyig fields E ˆ yˆ zˆ E K / / y / z y z z y z y ˆ + yˆ + zˆ y z z y ˆ yˆ zˆ (b) E K / / y / z y z ˆ yˆ zˆ (c) E K / / y / z y z y z y ˆ + yˆ + + zˆ y z z y y y ˆ + + yˆ + zˆ z z y fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 7

47 ˆ yˆ zˆ ( z) ( z) (d) E K / / y / z y z zˆ y y ˆ + + yˆ + zˆ y z z y Q6. A charged particle is released at time t, from the origi i the presece of uiform static electric ad magetic fields give by E E yˆ ad B B zˆ respectively. Which of the followig statemets is true fort >? The particle moves alog the -ais. (b) The particle moves i a circular orbit. (c) The particle moves i the (, y) plae. (d) Particle moves i the ( y, z) plae As: (c) Solutio: I a cycloid charged particle will always cofie i a plae perpedicular to B. JEST-4 Q7. For a optical fiber with core ad claddig ide of. 45 ad. 44, respectively, what is the approimate cut-off agle of icidece? Cut-off agle of icidece is defied as the icidece agle below which light will be guided. As.: o 7 (b) / o (c) Solutio: θ si where.44,. 45 o 5 (d) / θ si θ si (.76) o θ fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 8

48 Q8. Two large ocoductig sheets oe with a fied uiform positive charge ad aother with a fied uiform egative charge are placed at a distace of meter from each other. The magitude of the surface charge desities are σ + 6.8μ C / m for the positively charged sheet ad σ 4.μ C / m for the egatively charged sheet. What is the electric field i the regio betwee the sheets? N / C (b).84 N / C 5 5 (c).4 N / C (d).6 N / C As.: Solutio: Electric field betwee the sheet is σ+ σ N / C 8.86 Q9. A system of two circular co-aial coils carryig equal currets I alog same directio havig equal radius ad separated by a distace (as show i the figure below). The magitude of magetic field at the midpoit P is give by μ I 4μ (b) I I I 5 5 Y (c) As.: (c) 5 μ I 8 5 Solutio: B μ I ( + d ) B (d) μ I + 4, B μ I + 4 P d X B B + B μ I 5 4 I 4 8μ 5 5 μ B 5 I fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 9

49 Q. Fid the resoace frequecy (rad/sec) of the circuit show i the figure below V.5F Ω Ω H.4 (b). (c). (d).7 As.: (b) Solutio:. LC L (where Ω, L H, C.5 F ) Q. A electro is eecutig simple harmoic motio alog the y-ais i right haded coordiate system. Which of the followig statemets is true for emitted radiatio? The radiatio will be most itese i z plae (b) The radiatio will be most itese i y plae (c) The radiatio will violate causality (d) The electro s rest mass eergy will reduce due to radiatio loss As.: Solutio: Oscillatig electro does ot emit radiatio i the directio of oscillatio. I the perpedicular directio of oscillatio itesity is maimum. So i this case the itesity will be maimum alog ad z - ais or z - plae (itesity is also e y -plae but less). Q. A coductig sphere of radius r has charge Q o its surface. If the charge o the sphere is doubled ad its radius is halved, the eergy associated with the electric field will icrease four times (b) icrease eight times (c) remai the same (d) decrease four times As.: (b) Q Q Solutio: E rw ˆ E 4πrdr+ E4πrdr W 4π r 8π ( ) Q 8Q W 8W 8π 8π fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

50 JEST- Q. At equilibrium, there ca ot be ay free charge iside a metal. However, if you forcibly put charge i the iterior the it takes some fiite time to disappear i.e. move to the 6 surface. If the coductivity σ of a metal is ( ) ε 8.85 Farad/m, this time will be approimately: Ω m ad the dielectric costat 5 sec (b) sec (c) 9 sec (d) 7 sec As.: (d) Solutio: Characteristic time: τ σ Q4. The electric fields outside ( r > ) ad iside ( r ) volume charge desity are give by E r > < a solid sphere with a uiform 4πε q r rˆ ad E r < 4πε respectively, while the electric field outside a spherical shell with a uiform surface q charge desity is give by Er< rˆ, q beig the total charge. The correct ratio 4πε r of the electrostatic eergies for the secod case to the first case is : (b) 9:6 (c) :8 (d) 5:6 As.: (d) Solutio: Electrostatic eergy i spherical shell w sp q q q 4 r dr π 4 ( 4π ) r 8π r 8π E 4πr dr + E 4πr dr q rrˆ Electrostatic eergy i solid sphere w s E 4π r dr + E 4πr dr 5 q r q + 6 π 5 8π r 8 q q 6q + 5 8π 8π 4π w s fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

51 Now fiziks W W spherical sphere q 8π 5 6q 6 4π Q5. A thi uiform rig carryig charge Q ad mass M rotates about its ais. What is the As.: (c) gyromagetic ratio (defied as ratio of magetic dipole momet to the agular mometum) of this rig? Q /( πm ) (b) Q/M (c) Q/(M) (d) Q /( πm ) Solutio: Magetic dipole momet Q Q Q ω M IA πr π πr r T π T M Q Agular mometum J Mr ω J M Q6. The electric ad magetic field caused by a accelerated charged particle are foud to scale as E r ad B m r at large distaces. What are the value of ad m?, m (b), m (c), m (d), m As.: (c) qasiθ qasiθ Solutio: For large distace F, B E, B r r r r So m Q7. If E yiˆ + yzj ˆ zkˆ ad E y ˆ i + ( y + z ) ˆj yzkˆ the + Both are impossible electrostatic fields (b) Both are possible electrostatic fields + (c) Oly E is a possible electrostatic field (d) Oly E is a possible electrostatic field As.: (d) Solutio: For electrostatic field E fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com

52 fiziks iˆ ˆj kˆ y z E y y + z yz ( z z) iˆ + + ( y y) zˆ iˆ ˆj kˆ E ( y) iˆ+ + j ˆ y z y yz yz Q8. A charge q is placed at the cetre of a otherwise eutral dielectric sphere of radius a ad relative permittivityε r. We deote the epressio followig statemets is false? The electric field iside the sphere, r (b) The field outside the sphere, r (c) The total charge iside a sphere of radius r (d) The total charge iside a sphere of radius r fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / a q / 4πε r by ( ) E r <, is give by ( )/ r > a, is give by E( r ) E r. Which of the 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com ε > a is give by q. < a is give by q. As.: (d) Q9. A electromagetic wave of frequecy ω travels i the - directio through vacuum. It ω is polarized i the y - directio ad the amplitude of the electric field is E. With k c where c is the speed of light i vacuum, the electric ad the magetic fields are the covetioally give by E E E cos( ky t)ˆ ω ad B cos( ky ω t)zˆ c E (b) E E cos( k t)yˆ ω ad B cos( k ω t)zˆ c E (c) E E cos( k ω t)zˆ ad B cos( ky ω t)yˆ c E (d) E E cos( k t)ˆ ω ad B cos( ky ω t)yˆ c

53 As.: (b) E E cos k ωt y Solutio: ( ) ˆ B ( kˆ E) B ˆ E cos( k ωt) yˆ c c E E c c fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / ( ω )( ˆ ˆ) cos( ω )( ˆ) B cos k t y B k t z Q. A magetic field B B ( iˆ ˆj 4kˆ + ) velocity, v v ( ˆ i ˆj kˆ ) + that poit i SI uits is E v B ˆ i ˆj ( 4kˆ ) (c) E v B ( 4 ˆj 7kˆ ) + As. : (d) Solutio: F q[ E + v B] E ( v B) E v B 4 4 iˆ+ + ˆj+ 6+ kˆ JEST- eists at poit. If a test charge movig with a eperieces o force at a certai poit, the electric field at (b) E v B ( iˆ ˆj 7kˆ + + ) (d) E v B ( 4 ˆj 7kˆ ) v B + ( 4 ˆj 7kˆ ) + Q. A observer i a iertial frame fids that at a poit P the electric field vaishes but the magetic field does ot. This implies that i ay other iertial frame the electric field E ad the magetic field B satisfy E B (b) E B (c) E B (d) E As.: (b) Q. A circular coductig rig of radius rotates with costat agular velocity ω about its diameter placed alog the -ais. A uiform magetic field B is applied alog the y-ais. If at time t the rig is etirely i the y-plae, the emf iduced i the rig at time t > is {( ) ( ) ( ) } Bω π t (b) Bωπ ta ( ω t) (c) Bωπ si ( ω t) (d) Bωπ cos ( ω t) 8 B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 4

54 As. : (c) Solutio: φ B A BAcos θ BAcosω t ε m dφ d d dt dt dt m ε Bπ ωsiωt ( B A) [ BAcosωt] ε Bωπ siωt E BA( siω t)ω Q. A electric field i a regio is give by (, y, z) ai + czj + 6byk ˆ ˆ ˆ. For which values of a, b, c does this represet a electrostatic field?,, (b) 7, 6, (c),, 6 (d) 45, 6, As.: (c) Solutio: For electrostatic field E iˆ ˆj kˆ E E ( 6b c) iˆ+ ˆj y z [ ] + kˆ [ ] a cz 6by ( 6b c)ˆ i c 6b Q4. A capacitor C is coected to a battery V through three equal resistors ad a switch S as show below: S V C As.: (d) The capacitor is iitially ucharged. At timet, the switch S is closed. The voltage across the capacitor as a fuctio of time t for t > is give by ( /)( ep ( / )) (b) ( )( ( )) / ep / (c) ( /)( ep( / )) (d) ( )( ( )) / ep / fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 5

55 Solutio: S + + V i i + i i C Apply KVL i loop : V + i ( t) + ( i i ) ( ) ( ) Apply KVL i loop : t i + idt C ( i i ) V + C t idt i (ii) C di dt i + Iitial Coditios i + ( ) V V V V + ( ) i V + i + C t idt i i t i t V.. (i) di i dt di i i () t Ke C dt C V V K i () t e V V + t C + + ad i ( ), v C ( ) t C V vc () t i dt e dt C t t t C C t t C V e C C t V C C vc () t e C V vc () t e t C fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 6

56 Q5. A small maget is dropped dow a log vertical copper tube i a uiform gravitatioal field. After a log time, the maget attais a costat velocity (b) moves with a costat acceleratio (c) moves with a costat deceleratio (d) eecutes simple harmoic motio As. : Q6. Cosider a particle of electric charge e ad mass m movig uder the ifluece of a costat horizotal electric field E ad costat vertical gravitatioal field described by acceleratio due to gravity g. If the particle starts from rest, what will be its trajectory? parabolic (b) elliptic (c) straight lie (d) circular As.: (c) md d Solutio: Equatio of motio qe α t + c dt dt qe at t, v, α m similarly, mg md y dt α t α y y, α g α d α t αt dt Q7. A poit charge +q is placed at (,, d) above a grouded ifiite coductig plae defied by z. There are o charges preset aywhere else. What is the magitude of electric field at (,, - d)? As.: (d) q /( 8π d ) (b) - (c) (d) q /( 6π d ) Solutio: Electric field at Q q E 4 π q E 6π d ( ) ( z ˆ ) d q 6π d zˆ fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 7

57 Q8. A time-depedet magetic field B ( t) is produced i a circular regio of space, ifiitely log ad of radius. The magetic field is give as B B t zˆ ad is zero for r >, where B is a positive costat. The electric field at poit r is B B r ˆ (b) ˆ B θ (c) B r ˆ (d) ˆ θ 4 4 As.: (b) B Solutio: Solutio: E dl da E π r Bπ t lie B E B E θˆ r r B The electric field at poit r is E ˆ θ 4 Q9. Whe upolarised light is icidet o a glass plate at a particular agle, it is observed that the reflected beam is liearly polarized. What is the agle of the refracted beam with respect to the surface ormal? 56.7 (b).4 (c). (d) The light is completely reflected ad there is o refracted beam. As.: (b) Solutio: Sice,. 5 Brewster agle θ B Now θ fiziks, H.No. 4 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi 6 Phoe: / ta ta Q4. A cube has a costat electric potetial V o its surface. If there are o charges iside the cube, the potetial at the ceter of the cube is V (b) V / 8 (c) (d) V / 6 As.: B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi 6 fiziks.physics@gmail.com 8