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Physics Solution CPT_2 Date: 27-4-2014

1. Two charges each equal to )3( 1 q are placed at the corners of an equilateral triangle of side a. The electric field at the third corner is 3E where )4/(

200 aqE

(a) 03 EE (b) 03 EE (c) 03 EE (d) 03 EE Sol. (c) .

4,

4 2022

01

a

qE

a

qE Therefore 21 EEE

.4

360cos2

20

2122

21

a

qEEEE o

Since .13,31,31

20

20 44

3

a

q

a

q 20003 4 aqEEE . 2. Four charges are arranged at the corners of a square ABCD , as shown in the adjoining figure. The force

on the charge kept at the centre O is

(a) Zero (b) Along the diagonal AC

(c) Along the diagonal BD (d) Perpendicular to side AB

Sol. (c) We put a unit positive charge at O. Resultant force due to the charge placed at A and C is zero and

resultant charge due to B and D is towards D along the diagonal BD.

3. Two identical balls having like charges and placed at a certain distance apart repel each other with a

certain force. They are brought in contact and then moved apart to a distance equal to half their initial

separation. The force of repulsion between them increases 4.5 times in comparison with the initial

value. The ratio of the initial charges of the balls is

(a) 2 (b) 3 (c) 4 (d) 6

Sol. (a) Suppose the balls having charges Q1 and Q2 respectively.

Initially :

Finally :

2

221

2

2

21

)(

2

2'

r

QQk

r

QQk

F

It is given that FF 5.4' so

221

2

221 .5.4)(

r

QQk

r

QQk 21221 5.4)( QQQQ . On solving it gives .

1

2

2

1 Q

Q

4. C2 and C6 two charges are repelling each other with a force of N12 . If each charge is given C2 of charge, then the value of the force will be

(a) N4 (Attractive) (b) N4 (Repulsive) (c) N8 (Repulsive) (d) Zero

O

C

+q

+2q

B A +q

D

2q

A B

r

Q1 Q2

r

221 QQ

2

21 QQ A B

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Sol. (d) Resultant charges after adding the 2C be 0)22( and C4)62( F0

4022

21 r

kr

QQk

5. Three equal charges are placed on the three corners of a square. If the force between 1q and 2q is 12F

and that between 1q and 3q is 13F , the ratio of magnitudes 13

12

F

F is

(a) 2/1 (b) 2 (c) 2/1 (d) 2

sol. (b) 2

2

012

4

1

a

qF and 22013 )2(4 1 a qF 21312 FF

6. An electron falls through a small distance in a uniform electric field of magnitude 14102 NC . The direction of the field is reversed keeping the magnitude unchanged and a proton falls through the same

distance. The time of fall will be

(a) Same in both cases

(b) More in the case of an electron

(c) More in the case of proton

(d) Independent of charge

Sol. (c) The time required to fall through distance d is 22

1t

m

qEd or qEdmt 2

Since ,2 mt a proton takes more time. 7. Two similar spheres having q and q charge are kept at a certain distance. F force acts between the

two. If in the middle of two spheres, another similar sphere having q charge is kept, then it experience a force in magnitude and direction as

(a) Zero having no direction (b) F8 towards q charge

(c) F8 towards q charge (d) F4 towards q charge Sol. (c) Initially, force between A and C

2

2

r

QkF

When a similar sphere B having charge +Q is kept at the mid point of line joining A and C, then Net

force on B is CAnet FFF FrkQrkQrQk 8822 222222 . (Direction is shown in figure)

8. Electric charges of CC 1,1 and C2 are placed in air at the corners A, B and C respectively of an equilateral triangle ABC having length of each side 10 cm. The resultant force on the charge at C is

(a) 0.9 N (b) 1.8 N (c) 2.7 N (d) 3.6 N

Sol. (b) FA = force on C due to charge placed at A

N8.1)1010(

10210109

22

669

FB = force on C due to charge placed at B

N8.1)1.0(

10210109

2

669

r

A C

Q +Q B

r/2 r/2

+Q

FA FC

+1C 1C

+2C

10 cm

FB

B A

C

FA

120o

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Net force on C

NFFFFF oBABAnet 8.1120cos2)()(22

9. Two particle of equal mass m and charge q are placed at a distance of 16 cm. They do not experience

any force. The value of m

q is

(a) l (b) G

0 (c) 04G (d) G04

Sol. (d) They will not experience any force if |||| eG FF

22

2

022

2

)1016(.

4

1

)1016( qmG Gmq 04 10. Three charges each of magnitude q are placed at the corners of an equilateral triangle, the electrostatic

force on the charge placed at the center is (each side of triangle is L)

(a) Zero (b) 2

2

04

1

L

q (c) 220 34 1 Lq (d) 220121 Lq sol. (a) In the following figure since |||||| CBA FFF and they are equally inclined with each other, so

their resultant will be zero.

11. Two point charges placed at a certain distance r in air exert a force F on each other. Then the distance r'

at which these charges will exert the same force in a medium of dielectric constant k is given by

(a) r (b) r/k (c) kr / (d) kr

sol. (c) 'FF or K

rr

Kr

QQ

r

QQ ''44 20

212

0

21

12. Five balls numbered 1 to 5 are suspended using separate threads. Pairs (1, 2), (2, 4) and (4, 1) show

electrostatic attraction, while pair (2, 3) and (4, 5) show repulsion. Therefore ball 1 must be

(a) Positively charged (b) Negatively charged (c) Neutral (d) Made of metal

sol. (c) Let us consider 1 ball has any type of charge. 1 and 2 must have different charges, 2 and 4 must

have different charges i.e. 1 and 4 must have same charges but electrostatics attraction is also present in

(1, 4) which is impossible.

13. Equal charges q are placed at the four corners DCBA ,,, of a square of length a . The magnitude of the

force on the charge at B will be

(a) 2

0

2

4

3

a

q (b) 202

4

4

a

q (c) 202

42

221

a

q (d) 202

42

12

a

q SOL. (c) After following the guidelines mentioned above

q A

B C

q q FA

Q FC FB

A

D C

B

FC

FAC FA

FD

+Q +Q

+Q

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DCADACnet FFFFFF 22 Since

2

2

a

kqFF CA and 22)2(akqFD

21222 22222 2 akqakqakqFnet

2

221

4 20

2

a

q

14. A charge q is placed at the centre of the line joining two equal charges Q. The system of the three

charges will be in equilibrium, if q is equal to

(a) 2

Q (b) 4

Q (c) 4

Q (d) 2

Q

Sol. (b) Suppose in the following figure, equilibrium of charge B is considered. Hence for it's equilibrium |||| CA FF

2

2

0 44

1

x

Q 204 1 xqQ 4Qq

Short Trick : For such type of problem the magnitude of middle charge can be determined if either of

the extreme charge is in equilibrium by using the following formula.

If charge A is in equilibrium then q = 2

1 xxQB If charge B is in equilibrium then

2

2 xxQq A If the whole system is in equilibrium then use either of the above formula.

15. ABC is an equilateral triangle. Charges q are placed at each corner. The electric intensity at O will be

(a) 2

04

1

r

q (b)

r

q

04

1 (c) Zero

(d) 2

0

3

4

1

r

q sol. (c)

16. The magnitude of electric field intensity E is such that, an electron placed in it would experience an

electrical force equal to its weight is given by

+q +q

+q

r

r r

A

B C

O

QA = Q q

x

x1 x2

FC FA

A B C

QB = Q

EC =E EB =E

EA = E

120o

120o

120o

EA = E

EBC = E

EC EB

EA

Enet = 0

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(a) mge (b) e

mg (c)

mg

e (d) g

m

e2

2

SOL. (b) According to the question, mgeE

e

mgE

17. An uncharged sphere of metal is placed in between two charged plates as shown. The lines of force

look like

(a) A (b) B (c) C (d) D

sol. (c) Electric lines of force never intersect the conductor. They are perpendicular and slightly curved

near the surface of conductor.

18. The intensity of electric field required to balance a proton of mass kg27107.1 and charge C19106.1 is nearly

(a) mV /101 7 (b) mV /101 5 (c) mV /101 7 (d) mV /101 5

SOL. (a) Since qE = mg or 19

27

106.1

8.9107.1 qmgE

= 10.0 108 = 1 107 V/m 19. Two point charges Q and 3Q are placed at some distance apart. If the electric field at the location of

Q is E then at the locality of Q3 , it is (a) E (b) 3/E (c) E3 (d) 3/E sol. (b) The field produced by charge 3Q at A, this is E as mentioned in the Example.

2

3

x

QE (along AB directed towards negative charge)

Now field at location of 3Q i.e. field at B due to charge Q will be 32E

x

QE' (along AB directed

away from positive charge)

20. The distance between a proton and electron both having a charge coulomb19106.1 , of a hydrogen atom is metre1010 . The value of intensity of electric field produced on electron due to proton will be

(a) CN /10304.2 10 (b) mV /4.14 (c) mV /16 (d) CN /1044.1 11 SOL. (d) 2

04 r

qE CN /1044.1)10( 106.1109 11210 199

21. Two positive charges of 20 coulomb and coulombQ are situated at a distance of cm60 . The neutral point

between them is at a distance of cm20 from the coulomb20 charge. Charge Q is

(a) C30 (b) C40 (c) C60 (d) C80

sol. (d) At neutral point

+ + + + + + +

A

+ + + + + + +

B

D

+ + + + + + +

C

+ + + + + + +

A B

x

Q 3 Q

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2222 )1040()1020(

20 Qkk Q = 80 C 22. Two equal negative charge q are fixed at the fixed points ),0( a and ),0( a on the Y-axis. A positive

charge Q is released from rest at the point )0,2( a on the X-axis. The charge Q will

(a) Execute simple harmonic motion about the origin

(b) Move to the origin and remain at rest

(c) Move to infinity

(d) Execute oscillatory but not simple harmonic motion

sol (d) By symmetry of problem the components of force on Q due to charges at A and B along y-axis will

cancel each other while along x-axis will add up and will be along CO. Under the action of this force

charge Q will move towards O. If at any time charge Q is at a distance x from O. Net force on charge Q

212222

0 )()(4

12cos2

xa

x

xa

qQFFnet

i.e., 23220 2.4 1 xa qQxFnet

As the restoring force Fnet is not linear, motion will be oscillatory (with amplitude 2a) but not simple

harmonic.

23. A positively charged ball hangs from a silk thread. We put a positive test charge 0q at a point and

measure 0/ qF , then it can be predicted that the electric field strength E

(a) 0/ qF (b) 0/ qF (c) 0/ qF (d) Cannot be estimated Sol. (a) Because of the presence of positive test charge q0 in front of positively charged ball, charge on the

ball will be redistributed, less charge on the front half surface and more charge on the back half surface.

As a result of this net force F between ball and point charge will decrease i.e. actual electric field will

be greater than ./ 0qF

24. In the given figure two tiny conducting balls of identical mass m and identical charge q hang from non-

conducting threads of equal length L. Assume that is so small that sintan , then for equilibrium x is equal to

(a) 3

1

0

2

2 mgLq (b) 31

0

2

2 mgqL (c) 31

0

22

4 mgLq (d) 31

0

2

4 mgLq sol. (a)

In equilibrium Fe = T sin ....... (i)

2a

x

O

C

A q

B

a

a

q

Q

F

F

x

T sin T Fe

mg

T cos

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mg = T cos ....... (ii)

mgx

q

mg

F

o

e 224tan also Lx 2/sintan Hence

mgx

q

L

x

o 22

42

mg

Lqx

o42 23 3/122 mgLqx o 25. Two equal charges are separated by a distance d. A third charge placed on a perpendicular bisector at x

distance will experience maximum coulomb force when

(a) 2

dx (b)

2

dx (c)

22

dx (d)

32

dx

Sol. (c) Suppose third charge is similar to Q and it is q

So net force on it

Fnet = 2F cos

Where

4

.4

12

20 dx

QqF and

4

cos2

2 dx

x

2/12

2

220

44

.4

12

dx

x

dx

QqFnet

2/3

22

04

4

2

dx

Qqx

For Fnet to be maximum 0

dx

dFnet

i.e. 0

44

22/3

22

0

dx

Qqx

dx

d

or 04

34

2/52

22

2/32

2

dxx

dx

i.e. 22

dx

26. A small sphere carrying a charge q is hanging in between two parallel plates by a string of length L. Time period of pendulum is 0T . When parallel plates are charged, the time period changes to T . The

ratio 0/ TT is equal to

4/22

dx 4/22 dx

B C

Q Q

F F q

x

2

d

2

d

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(a)

2/1

gm

qEg

(b)

2/3

m

qEg

g (c)

2/1

m

qEg

g (d) None of these

Sol. (c)

Net downward force QEmgmg ' Effect acceleration mQEgg' Hence time period

m

QEg

l

g

lT 2

'2

27. Three charges 21, qq and 3q are placed as shown in the figure. The x-component of the force on 1q is proportional to

(a) sin23

22

a

q

b

q (b) cos

23

22

a

q

b

q (c) sin

23

22

a

q

b

q (d) cos

23

22

a

q

b

q

sol. (c)

F2 = Force applied by 2q on 1q F3 = Force applied by )( 3q on 1q x-component of Net force on 1q is Fx = F2 + F3 sin sin. 231221 aqqkbqqk sin231221 aqqbqqkFx sin23221 aqbqqkFx sin2322 aqbqFx

28. Six charges, three positive and three negative of equal magnitude are to be placed at the vertices of a

regular hexagon such that the electric field at O is double the electric field when only one positive

charge of same magnitude is placed at R. Which of the following arrangements of charges is possible

for P, Q, R, S, T and U respectively

(a) ,,,,,

b

a

Y

X

q3

+q2 q1

P Q

R U

T S

O

b

a

F3 cos F3 F2

+q2

q3 F3 sin q1

E

QE

mg

+

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(b) ,,,,, (c) ,,,,, (d) ,,,,, sol. (d) If the charges are arranged according to the option (d), the electric fields due to P and S and due to

Q and T add to zero, while due to U and R will be added up.

29. Two identical point charges are placed at a separation of d. P is a point on the line joining the charges,

at a distance x from any one charge. The field at P is E, E is plotted against x for values of x from close

to zero to slightly less than d. Which of the following represents the resulting curve

(a) (b)

(c) (d)

Sol. (d) At mid point, E = 0

Before mid point, E is positive. This is maximum near the charge and decreases towards mid point.

After mid point, E is negative, The curve crosses x-axes at x-axis at x = d/2. From centre to end, E

decreases.

The variation is shown by curve.

30. Assertion : Charge is quantized

Reason : Charge, which is less than 1 C is not possible

(a) If both assertion and reason are true and the reason is the correct explanation of the assertion.

(b) If both assertion and reason are true but reason is not the correct explanation of the assertion.

(c) If assertion is true but reason is false.

(d) If assertion is false but reason is true.

Sol. (c) neQ and charge lesser than 1 C is possible.

Y

X O

E

x

Y

X O

E

x

Y

X O

E

x

Y

X O

E

x

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