DPP No. # 4 PART–I

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DATE : 25.04.2019 ADVANCED PART TEST (APT)–2

Syllabus : Current electricity, Capacitor, Error Analysis, Magnetic field and force, Electromagnetic induction, Alternating Current.

TARGET : JEE (ADVANCED) 2019

EE ST

INFORMA TIO

Course : VIJETA & VIJAY (ADP & ADR) Date : 20.04.2019

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DPPDPPDPPDAILY PRACTICE PROBLEMS

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NO. 4

DPP Syllabus : Magnetic field and force, Electromagnetic Induction, Alternating Current

DPP No. # 4

PART–I Total Marks : 85 Max. Time : 75 min. Single choice Objective (-1 negative marking) Q. 1 to 5 (3 marks 3 min.) [15, 15] Multiple choice objective (–2 negative & partial marking) Q.6 to 13 (4 marks, 3 min.) [32, 24] Single Digit Subjective Questions (-1 negative marking) Q.14 to Q.18 (3 marks, 3 min.) [15, 15] Comprehension (MCQ) ('–2' negative & Partial marking) Q.19 to Q.20 (4 marks 3 min.) [08, 06] Comprehension (-1 negative marking) Q.21 to 24 (3 marks 3 min.) [12, 12] Match Listing (-1 negative marking) Q.25 (3 marks, 3 min.) [03, 03]

1. In the given circuit Key is closed at t = 0. At what time the P.D across inductor is one fourth of emf of the

cell.

(A) L

n2R (B)

Ln4

R (C) 0 (D)

2L

R

2. Conductor of length has shape of a semi cylinder of radius R (<< ). Cross section of the conductor is

shown in the figure. Thickness of the conductor is t (<< R) and conductivity of its material varies with angle according to the law = 0 cos where 0 is a constant. If a battery of emf is connected across its end faces (across the semi–circular cross-sections), the magnetic induction at the mid point O of the axis of the semi-cylinder is :

(A) 0 0 t

8

(B) 0 0 t

4

(C) 0 0 t

(D) 0 02 t




3. In the ac circuit shown, XL=7 ,R=4 and X

C=4 .The reading of the ideal voltmeter V

2 is 8 2 . The

reading of the ideal ammeter:

(A) 3A (B) 2A (C) 4A (D) 5A 4. Consider a square frame of side 1m and total resistance 16 (uniformly

distributed) in two regions of uniform magnetic field. In region left to line AC magnetic field is 200 T inward and perpendicular to the plane. In region right to the line AC magnetic field is 200 T outward and perpendicular to the plane. A conducting rod of negligible resistance slides with uniform velocity 0.5 m/s as shown in the figure. The rod always remain parallel to side BC. Current through the conducting rod when it is 25 cm away from side BC is

×

A B

C D × ×

× ×

×

(A) zero (B) A3

20 (C) A

3

40 (D) 10A

5. Three identical large plates are fixed at separation of d from each other

as shown. The area of each plate is A. Plate 1 is given charge +Q while plates 2 and 3 are neutral and are connected to each other through coil of inductances L and switch S. If resistance of all connected wires is neglected the maximum current flow through coil after closing switch is (C = 0 A/d) (neglect fringe effect)

(A) LC

Q0 (B) LC2

Q0 (C) LC

Q2 0 (D) LC2

Q0

6.

In the circuit shown, resistance R = 100 , inductance L = 2

H

and capacitance C = 8

F

are connected in series with an ac

source of 200 volt and frequency ‘f’. If the readings of the hot wire voltmeters V

1 and V

2 are same then :

(A) f = 125 Hz (B) f = 250 Hz (C) current through R is 2A (D) V

1 = V

2 = 1000 volt

7. Consider a series LCR circuit connected to an AC supply of 220 V. If voltage drop across resistance R

is VR, voltage drop across capacitor is Vc = 2VR and that across inductor coil is VL = 3VR then choose correct alternative(s)

(A) V2220VR

(B) Power factor of circuit is 2

1

(C) VR = 156 V

(D) Phase difference between current and source voltage is 4

S

L

Q d 21 3d




8. A proton is fired from origin with velocity 0 0ˆ ˆv v j v k

in a uniform magnetic field 0

ˆB B j

In the subsequent motion of the proton (v0 and B0 are positive) (A) its z-coordinate can never be negative (B) its x-coordinate can never be positive (C) its x-and z-coordinates cannot be zero at the same time (D) its y-coordinate will be proportional to its time of flight 9. A wooden cubical block ABCDEFGO of mass m and side 'a'

is wrapped by a square wire loop of perimeter 4a at z = a/2 as shown. The current in the wire I. The whole system is placed on frictionless horizontal surface in a uniform

magnetic field 0ˆB B j

In this situation. Line of action of

normal force between horizontal surface and block passes through a point at a distance x from centre. Select the correct statement(s)

E

B

y

a

C

G

B

A x

D

O

z

F

(A) The block must not topple if 0

mgI

aB (B) The block must not topple if

0

mgI

2aB

(C) a

x4

if 0

mgI

2aB (D)

ax

4 if

0

mgI

4aB

10. A capacitor with charge Q on it is connected to an inductor L as shown in diagram at t = 0. When the switch is flipped from position 1 to 2, the current in the circuit is observed to be at half of its maximum value. Then

L L

1

C

2

(A) Charge on capacitor at the time was Q/2 (B) Charge on capacitor at that time was Q 3

2

(C) Total energy lost is 2Q

4C (D) Total energy lost is

2Q

8C

11. A ring of mass m and radius R is set into pure rolling on horizontal

rough surface, in a uniform magnetic field of strength B-as shown in the figure. A point charge of negligible mass is attached to rolling ring. Friction is sufficient so that it does not slip at any point of its motion. (is measured in clockwise from positive y-axis) (A) Ring will continue to move with constant velocity. (B) The value of friction acting on ring is Bqv cos . (C) The value of friction acting on ring is Bqv sin .

(D) Ring will lose contact with ground if v is greater than mg

2qB

× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×

m

y

v = /R

q

R v x




12. In the figure shown the key is switched on at t = 0. Let 1 and

2 be the

currents through inductors having self inductances L1 & L

2 at any time t

respectively. The magnetic energy stored in the inductors 1 and 2 be U1

and U2. Then 1

2

U

U at any instant of time is :

(A) 1

2

L

L (B) 2

1

L

L

(C) 1

2

(D) 2

1

13. There exist a uniform magnetic field perpendicular to plane of

the paper (inward) right side of the line AB. A particle of charge

–q and mass m enters a uniform magnetic field B

(perpendicular to paper inward) at P with a velocity V0 at an angle and leaves the field at Q with velocity V at angle as shown in figure. (A) = (B) V = V0

(C) PQ = 02mV sin

Bq

(D) The particle remains in the field for time t = 2m( )

Bq

X X

X

X

X

X

X

X

X

X

X

X

X

X

X

B

P

V

Q

V0

A

B 14. In the circuit diagram shown, XC = 100 , XL = 200 and

R = 100 . The effective current through the source is Xthen find out value of X :

15. There exist a uniform inward magnetic field B =1T in cylindrical region of radius r = 2 m. Outside the cylindrical region the magnetic field is outward but of same magnitude 1T. A point charge (q = 1C and m = 1 kg) is projected from a point A on circumference with velocity 2m/s towards center as shown in figure. After time t = N sec particle again passes through initial point A. Find the minimum possible value of N.

× × ×

× × × ×

× × × ×

× × ×

AO

16. Solid uniform conducting sphere of mass ‘m’ and charge Q, rotates about its axis of symmetry with

constant angular velocity ‘’ then the ratio of magnetic moment to the moment of inertia of the sphere is

m6

.xQ then x is : ( Neglect induced charges due to centrifugal force)

17. The current in the outer coil is varying with time as I = 2t2. The

heat developed in the inner coil between t = 0 to t seconds is

2 2 4 30

2

k a t

3b R, where the resistance of the inner coil is R and

b >> a. Find k.,

a

b

L1 L2

K

R

V1 2




18. Consider 3 identical conducting loops kept parallel to each other such that their axes coincide. Loop (1), loop (2) and loop (3) self inductance of each loop is L. Current in loop (1) is I1, in loop (2) I2 and in loop

(3) I3, rate of change in current in them at certain instant is I1d

dt,

I2d

dt,

I3d

dt respectively . Their mutual

inductance is Mij (i.e. for 1 & 2 its M12) current in loop (1) and (3) is in same sense and in (2) its in opposite sense :

d d

(1)

(2)

(3)

1 2 3

Given that if II I 31 2 dd d

2dt dt dt

, emf induced in loop (2) is zero. Find 12

L

M

Comprehension-1 A thin non conducting ring of mass m, radius R, carrying uniformly distributed charge q is placed on

smooth horizontal plane. There exist an uniform time varying magnetic field in a cylindrical region directed

vertically upward. Magnitude of magnetic field varies with time as B = B0 nt tesla, where n is a number. Centre of ring coincides with centre of cylindrical region. Ring was at rest at t = 0. Neglect the magnetic field produced due to any kind of motion of ring. Answer the following 2 questions.

19. Choose the correct option(s) regarding magnitude of induced electric field at the periphery of the ring.

(A)

E

t(if n = 1)

(B)

E

t(if n = 2)

(C)

E

t(if n = 2)

(D)

E

t(if n = 3)

(Parabolic)

20. Choose the correct option(s) regarding instantaneous power P delivered to the ring by the source of magnetic field.

(A) if n = 1, P = 2 2 20B q R

t4m

(B) if n = 2, P = 2 2 2

30B q Rt

4m

(C) if n = 4, P = 2 2 2

70B q Rt

m (D) if n = 3, P =

2 2 250B q R

t4m

Comprehension-2 Suppose E

and B

are electric field and magnetic field at a

point in space. The vector 0

E B

is known as the poynting

vector. The magnitude of poynting vector is the rate of flow of energy per unit area per unit time. The rate at which electromagnetic energy flows through any area is the flux ofpoynting vector through that area.

Consider a cylindrical conductor of length and radius of cross-section r.It carries current distributed

uniformly through out it’s cross section. Resistivity of conductor is .




21. What is magnitude of poynting vector at a point on lateral surface of the conductor ?

(A) 22 r

(B)2

32 r

(C) 2

2 22 r

(D) 2

2 32 r

22. Total energy radiated per unit time through half cylindrical surface of the conductor is –

(A) 2

2r

(B) 2

2 2r

(C) 2

22 r

(D) Zero

Comprehension-3

Analysis of an alternating current circuit becomes easier if complex numbers are used to represent impedance as described below.

The resistance (R), inductive reactance (L) and capacitive reactance 1

C

are represented by the

numbers R, jL , – j1

C

respectively.

[Here j = 1 : Multiplication by j shows potential difference leads current by 2

and multiplication

by –j shows potential difference lags behind the current by 2

].

The effective complex impedance Z is obtained by grouping R, jL and –j 1

C

just as in the case of

grouping of resistance in direct current circuit. Now |Z| gives the true value of the impedance and arg Z = gives the phase angle by which emf is ahead of the current. If I is the R.M.S value of the current through the combinations and E the R.M.S value of emf across it, they are related as

E = |Z| I 23. The complex impedance of a device in an alternating current circuit is (4 + 3j) ohm. When the current

through the device is 2A, the potential difference across it and the phase angle by which p.d leads

current are respectively. (A) 10 V, 370 (B) 10 V, 530 (C) 50 V, 0 (D) 14 V, 0 24. In the circuit, the value of the impedance is

(A) 2

2 1R L

C

(B) 1

R LC

(C) 1

R LC

(D) 2 2

22 2

LR

(1 LC )




25. The figures in list- show some charge and current distribution with a charged particle projected in some specific direction list- gives certain conditions which may exist in the subsequent motion of the charged particle. Match list with list and select the correct answer using the codes given below the lists : (Neglect gravity force on charged particle)

List -I List-II

(P) A positive charge projected along the axis of two (1) Magnitude of acceleration of coaxial current carrying loops, carrying currents the charged particle is in opposite directions as shown constant

1 2

v + q

(Q) Two long line charges having equal charge density (2) Kinetic energy of the charged parallel to z-axis passing through points A (a,0,0) particle is constant and B (-a,0,0) and a positive charge projected along Y-axis from point C(0,2a,0)

Z

+ q

X

A

C B

Y

(R) A positive charge is projected from an inside point (3) Angular momentum of the on the axis of a long solenoid at some angle charged particle about the with axis as shown. Charge does not hit the solenoid point of projection is constant

+ q

(S) A positive charge projected from a point in between (4) Path of the charged particle two large parallel oppositely charged horizontal plates is straight at some angle with horizontal as shown. Charge does not hit the plate.

+ q

– – – – – – – – – – – –

+ + + + + + + + + + + +

Codes :

P Q R S (A) 3 1 2 4 (B) 3 2 1 4 (C) 4 2 1 3

(D) 3 4 2 1




PART–II Total Marks : 50 Max. Time : 45 min. Single choice Objective (-1 negative marking) Q. 1 to 5 (3 marks 3 min.) [15, 15] Multiple choice objective (-2 negative & partial marking) Q.6 to 10 (4 marks, 3 min.) [20, 15] Single Digit Subjective Questions (-1 negative marking) Q.11 to Q.14 (3 marks, 3 min.) [12, 12] Double Digits Subjective Questions (-1 negative marking) Q. 15 (3 marks, 3 min.) [03, 03]

1. In the first AC circuit current from the AC source is 1 = 0 sin (t –37°), in the second AC circuit current from AC source is I2 = I0 sin (t + 53°), current from AC source in third AC circuit will be :

Z1

N

E = E0 sint

Z2

N

E = E0 sint

Z1

N

Z2

E = E0 sint

(A) –10I 1sin t tan

72

(B) –10I 2sin t tan

72

(C) –10I 2sin t tan

2 7

(D) –10I 1sin t tan

2 7

2. A square loop of side length is placed near an infinitely long straight

current carrying wire. Current in the wire is . Distance of the nearer side of the loop from the wire is d. The wire and the loop are coplanar. Total resistance of the loop is R. If the loop translates through distance 2d in a direction shown in the figure, with the loop always remaining coplanar with wire then the charge flowing in it during the given displacement will be :

(A)

d

dn

R20 (B)

d2

d2n

R20

(C)

d2

dn

R20 (D)

d2

2d2n

R20

I

d

60° 2d

3. There exist a uniform magnetic field ˆB 10Tk

in space. A point charge (specific charge q

1C /kgm

is

projected from (–5 m, 0, 0) with velocity ˆ ˆ3vi vj . The value of 'v' for which the charge particle will pass

through origin. (A) 5m/s (B) 10m/s (C) 15m/s (D) 25m/s 4. Consider two uniformly charged concentric and coaxial rings of radii R and 2R. Total charge on inner ring

is Q1 and that on outer ring is Q2. Both rings are revolving in same sense with same angular velocity about its axis. If net magnetic induction at a distance R from the centre of the rings, on axis of rings is

zero then 2

1

Q

Q is :

(A) – 1 (B) 55

22 (C)

55

28 (D)

33

24




5. A thin wire bent into the form of a semicircle of radius r carries a current I. Magnetic field of induction at the point P as shown in the figure is:

(A) 0 n( 2 1)2 R

(B) 0 n( 2 1)

2 R

(C) 0 n( 2 1)4 R

(D) 0 n( 2 1)

4 R

R

P

6. Choose the correct options regarding the AC circuit given

in the figure: (Symbols have their usual meaning)

(A) Equivalent impedance of the circuit is 2R

3

(B) Power factor of the circuit is 3

2

(C) Average power for complete cycle is 203E

8R

(D) Effective reactance of the circuit is R

3

7. Consider a perfectly conducting uniform disc of mass m and radius 'a' hinged in vertical plane from its

centre and free to rotate with respect to hinge. A resistance R is connected between centre of the disc and periphery by using two sliding contacts C1 & C2. A long non conducting massless string is wrapped around the disc, whose another end is attached with a block of mass m. There exist a uniform horizontal magnetic field B. Whole arrangement is shown in the figure.

C1 C2

g

2mRm,R,

2

B

R

m Given system is released from rest at t = 0. Assume friction between string and disc is sufficient so that

there is no slipping between them. Let at any time t, velocity of block is v, angular velocity of disc is and current in resistance is i.

(A) From the energy equation mgv = 2dv dmv i R

dt dt

(B) The work done by the magnetic field is zero but it converts some part of the mechanical energy into heat.

(C) The velocity v, of the block as a function of time is tv (1 e )

(D) The acceleration of the block is tge

3




8. A nonconducting ring of uniform mass m, radius b and uniform linear charge density ‘’ is suspended as shown in figure in a gravity free space. There is uniform coaxial magnetic field B0, pointing up in a circular region of radius ‘a’ (<b). Now if this field is switched off, then :-

(A) There will be induced electric field on periphery of ring, in anticlockwise sense when seen from above

(B) Induced electric field imparts angular momentum of magnitude a2b Bo

(C) Final angular velocity of ring will be more if time taken to switch off the field (Bo) is small

(D) Final angular velocity will always be independent of time taken to switch off the field (Bo).

9. In the diagram shown, a uniform magnetic field is present perpendicular to the plane of the paper. Both

the rings are identical and have a constant resistance per unit length. The left ring has been kept fixed at its position and the right ring is slid uniformly on the left ring towards the right hand side. Which of the following statements is/are true ?

X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

(A) The emf induced in the left ring is zero (B) The emf induced in both the rings is non zero.. (C) The magnetic force acting on the right ring is zero. (D) The magnetic force acting on both the rings is non-zero. 10. A piece of copper foil is bent into the shape as illustrated in figure. The gap between two parallel planes

‘d’ is very small in comparison to rest dimensions. Current I which is uniformly distributed over width enters from edge AB and leaves at edge CD.

R

dBA

DC

a

Select correct alternatives

(A) The magnetic field at centre of cylinder is 0I

(B) The ratio of energy stored in cylindrical volume and energy stored in space between planes is =2R

ad

(C) The self inductance of the arrangement is 0

(R2 + ad)

(D) Total energy stored in arrangement is 2 2 2 20 ( R ad) I

2

a b

B0




11. Power factor of an L-R circuit is 0.6 and that of a C-R circuit is 36

37. If the elements [L, C and R] of the

two circuits are joined in series the power factor of the circuit is found to be 1. The ratio of resistance in the C-R circuit to the resistance in the L-R circuit is m : 1. Find the value of m.

12. In the shown circuit, R1 = 10 L =

3

10H, R2 = 20 , C =

3

2milli-farad

and t is time in seconds. Then at the instant current through R1 is 10 2

A ; find the current through resistor R2 in amperes.

13. Four identical charges are fixed on the periphery of a non conducting ring of radius R as shown. Ring is rotated with constant angular velocity w.r.t. an axis perpendicular to plane of the ring and passing through the centre of the ring. Force acting on any of the charge particle only due to

the magnetic interaction between the charges is F = 2 2

0q 1

4

Here is an integer. Find the value of .

+q

+q

+q

+q R

14. Figure shows a movable, rigid and conducting connector MN between a fixed L-shaped conducting wire. The whole system is in horizontal plane having vertical magnetic field B and point M is pulled horizontally with constant speed v starting from t = 0 by some external agent. Initially, at t = 0, if = 90°, induced current in the loop OMN is found to

be zero at time t = VK

. Find K (Given that M and N are always in

contact with the L–shaped wire).

V

M

N

O

B

15. The current density J

inside a long, solid, cylindrical wire of radius a = 12 mm is in the direction of the

central axis, and its magnitude varies linearly with radial distance r from the axis according to J = 0J r

a,

where J0 = 510

4A/m2. Find the magnitude of the magnetic field at r =

a

2 in µT.

C R2

R1L

V=200 2 sin(100t) volt

DPP No. # 4 PART–I

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