Electrostatics Exercise

E xercise1 A.1

Five ball s, numbered 1 to 5, are suspended suspended using separate threads. threads. Pairs (1, 2), (2, 4), (4, 1) show electrostatic attraction, while pairs (2, 3) and (4, 5) show repulsion, repulsion, therefore ball 1 must be: (A) Positi v ely charged (B) Negativ el y charged (C) Neutral (D) Made of metal

A.2*

Select the correct alternative : (A) The charge gained by the uncharged body from a charged body due to conduction is equal to half of the total charge initially present. (B) The magnitude of charge increases with the increase increase in v elocity of charge (C) Charge can not exist without matter although matter can ex ist without charge (D) Repulsion is the true test of electrification (electrification m eans body has net charge)

A.3

Mark out the correct option. (A) The total charge of the universe is constant. (B) The total positiv e charge of the universe is constant. (C) The total negative charge of the univ erse is constant constant (D) The total number of charged partic les in the universe is constant.

A.4

A poin t cha rge Q 1 exerts some force on a second point charge Q2. If a 3 rd point charge Q 3 is brought near, the force of Q 1 exerted on Q 2(Wit hout changing their respective positions): (A) W i l l increase (B) W il l dec rease (C) Will remain unchanged (D) Will increase if Q 3 is of the same sign as Q 1 and will decrease if Q3 is of opposite sign.

A.5

Three charge +4q, Q and q are placed in a straight line of length  at points distance 0, /2 and  respectively. What should be the value of Q in order to make the net f orce on q to be zero? ( A) – q (B) –2q (C) –q/2 (D) 4q

A.6

Two Two small bal ls having equal positive charge Q (coulomb) on each are suspended suspended by two insulating strings of equal length L, fro m a hook fix ed to a stand. If the whole set up is taken in a satellite then the angle  between the two strings strings is : (in equill ibrium) ( A) O º

A.7

( B) 9 0º

(D) 0 º <

 <180º

In above question the tension in each string is : (A) 0

A.8

(C) 180º

( B)

1

Q2

. 40 L2

(C)

1

Q2

. 40 2L2

(D)

1

Q2

. 40 4L2

Two Two point charges of the same m agnitude and opposite sign are f ixed at points A and B. A third point charge is to be balanced at point P by the electrostatic force due to these two charges. charges. The point P:

(A) (A) lie lies s on the the per perpe pend ndiicula cularr bis bisec ecto torr of of line line AB (C) l i es to t he l eft of A

(B) (B) is is at at the the mid mid poin pointt of lin line e AB (D) none of these.

A.9

The electric force on 2  c charge placed at the centre O of two equilateral triangles each of si de 10 cm, as shown in figure is P. If charge A, B, C, D, E & F are 2  c, 2  c, 2  c, -2 c, -2 - 2  c, -2 - 2  c respectivel y, then force acting on P is : ( A) 2 1. 6 N ( B) 64 . 8 N (C) 0 (D) 43.2 N

A.10

Two point charges q 1 = 20C and q2 = 25C are placed at (–1, 1, 1) m and (3, 1, –2)m, with respect respect to some coordinate axes. Find magnitude and unit vector alo ng force on q2?

A.11

What Wh at is the percentage change in distance if the force of attractio n between two two point charges increases to 4 tim es keeping magnitude of charges constant? constant?

A.12

Ten positively charged particles particles are kept fixed on the X-axis at points x = 10 cm, 20 cm, 30 cm, ....., 100 cm. The first particle has a charge 1.0 × 1.0 –8 C, the second 8 × 10 –8 C, the third 27 × 10 –8 C and so on. Find the magnitude of the electric force acting on a 1 C charge placed at the origin.

RESONANCE

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A.13

Two ch charge arged d pa partic rticle les s havi having ng cha charge 2.0 × 10 10 –8 C and mass 1.8 × 10 –6 Kg each are joined by an insulating string of length 1 m and the system is kept on a smooth horizontal table. Find the tension in the string. If sud sudde denly nly str strin ing g is cut cut the then n what what is the the acce acceler lerat atio ion n of each each bloc block? k? Are Are the they havi having ng equal ual acce accele lerratio ation n?

(i)

(ii) (ii) (iii iii) A.14

(i)

A charge q0 is placed at the centre of a regular pentagon havin g equal charges q at its corners. Find the force on q 0? (Assume a = distance from centre to corner for pentagon)

(ii) (ii) (iii) (iii)

If char charge ge of of one one of the the corn corner ers s is removed removed then then find find out magn magnit itude ude of force force acting acting on q0 ? If two two adjace adjacent nt charg charges es are removed from corners corners then then find find out out magnitud magnitude e of of force force acting acting on q0?

A.15

The distance distance between two fixed positive charges 4e and and e is  . How should a third charge ‘q’ be arranged for it to be in equilibrium? Under what condition will equilibrium of the charge ‘q’ be stable (for displacement on the line joining 4e and e) or will it be unstable?

A.16

Two particles A and B, each hav ing a charge Q are placed a distance d apart. W here should a particle of charge q be placed on the perpendicular bisector of AB AB so that it experiences maximum f orce? What is the magnitude of the maximum force?

B.1

There is a uniform electric field in X-direction. If the work done in mov ing a charge of 0.2 C through a distance of 2 metre along the line making an angle of 60º with X-direction is 4 joule, then the value of E is : (A)

B.2

( B) 4 N/ C

3 N/C

     2   (B)  g  qE   m  



g

     2   (C) g  qE   m 



2

 qE  g    m 

(D)

2

2

Three equal point charges +Q are present at the points A, B, C of a triangle having equal sides. The intensity intensity of electric fiel d at O will be: (A)

1

Q

(B)

40 r

2

(C) Zero

B.4

(D) 20 N/C

A simple pendulum has a length , mass of bob m. The bob is given a charge q coulomb. The pendulum is suspended suspended between the vertical pl ates of charged parallel plate capacitor. Which produces a uniform electric fi eld of strength E between the plates, then assuming that it does not collide with the plate calculate the time period T =

(A) 2

B.3

(C) 5 N/ C

(D)

1

Q

4 0 r

1

Q2

40 r 2

The maximum electric f ield intensity on the axis of a uniformly charged ring of charge q and radius radius R will be : (A)

1

q

(B)

1

2q

(C)

1

2q

(D)

1

3q









B.6

A charge ' q ' is placed at the centre of a conducti ng spherical shell of radius R, which is given a charge Q. An external charge Q  is also present at distance R (R > R) from ' q '. Then the resultant resultant fiel d will be best represented represented for region r < R by: [ where r is the distance of the point from q ]

(A)

B.7

B.9

(C)

(D)

In the above question if Q' is removed then which option is correct correct :

(A)

B.8

(B)

(B)

(C)

(D)

A solid soli d sphere of r adiu s R has a vo lume lu me cha rge  = 0 r 2 ( Where 0 is a constant and r is the distance from centre). At a distance distance x from its centre centre for x < R, the electric field is directly proportional to : (A) 1/x 2 ( B) 1/ x (C) x 3 (D) x 2 The linear charge density on upper half of a segment of ring is

 and at

lower half it is –  . The direction of electric field at centre O of ring is : (A) al ong OA ( B) al o ng O B (C) along OC (D) along OD B.10* B.10*

A large noncond nonconducting ucting sheet sheet M is given a uniform charge charge density density.. Two Two uncharged small m etal rods A and B are pl aced near the sheet as shown in figure. (A) M at tracts A ( B) M a t t r ac t s B (C) A at tracts B (D) B at tracts A

B.11* A unifo rm electric elect ric fiel d of strength E exists exi sts in a region. An electron elec tron (charge –e, mass m) enters a points A perpendi perp endi cularl cul arl y with velo v elocit city y V. V. It mov es through throug h the electri ele ctri c fie ld & exi ts at point B. Then: Then : 2 2amv ˆ i (A) E = 2 ed (B) Rate of work done by the electric fiel d at B is

4ma 2 v 3 d3

(C) Rate of work by the electric fiel d at A is zero (D) Velocity Velocity at B is

2av ˆ i d

 v jˆ

B.12* At distanc dist ance e of 5cm and 10cm outwar ds from fro m the surf ace of a unif orm ly char ged soli d s phere, pher e, the









B.13

If a positively charged pendulum is oscillating in a uniform electric field as shown in Figure. Its time peri od of SHM as compared to tha t when it was uncharged. (A) W i l l increase (B) W il l dec rease (C) W il l not change (D) W il l f irst increase then decrease

B.14

A water wate r par ticle tic le of m ass 10.0 10. 0 mg m g and an d hav ha v ing a charge cha rge of o f 1.50 × 10 1 0 –6 C stays suspended in a room. What is the magnitude of elec tric field in the room ? What is its direction ?

B.15

A unifor uni form m ele ctri c field fi eld E is crea ted between betwee n two parall para ll el, charged plates as shown in figure. An electron enters the fiel d symme tricall y between the plates with with a speed v0. The length of each plate is l . Find the angle of devi ation of the path of the electron as it comes out of the field.

B.16

Ten charges are placed on a meter stick at regular i nterval starting from 10cm mark at an i nterval of 10cm. The magnitude of the charges are q, 4q, 9q,.........Find the field intensity at the zero mark of the stick.

B.17

In the following figures find the electric fiel d at a point 'P' on the ax is of the square. The distance distance of 'P' from the centre is 'x'.

B.18

In the above question find electric field for two cases (a) x = 0 (b) x >> a. Also explain physical m eaning of result of part (b)

B.19

If three infinite charged sheets of uniform surface charge densities , 2 and –4 are placed as shown in figure, then find out el ectric field intensities at points A, B, C and D.

B.20

(i) (i)

(ii) (ii) B.21

The The elec electr tric ic field field int inten ensi sity ty at at a dist distan ance ce 20 20 cm fro from m the cent centre re of a uni unifor formly mly cha charg rged ed nonconducting solid sphere of radius 10 cm is E. Then what is the electric field intensity at a distance 5 cm from the centre? Find the electric field at r = 20 cm i n terms of E, and at 20 cm (outside) (outside) from the surface and at 5 cm (inside) from the surface. Ques Questi tion on if the the dist distan ance ces s give given n are are from from sur surfa face ce of sph spher ere. e.

Repeat the question if sphere is a hollow nonconducting sphere sphere of radius R and unifo rm surface charge density . (i) 0

R 2 ˆ r (i i )  0 (r  R )2

B.22

A charge of 16  10 -9 C is fixed at the origin of coor dinates, a second charge of unknown magnitude is at x = 3m, y = 0 and a third third charge of 12  10 -9 C is at x = 6m, y = 0. 0. What is the value of the the unknown unknown charge if the resultant resultant field at x = 8m, y = 0 is 20.25 N/C N/C directed directed towards towards positive positive x-axis?

B.23

A clock fac e has negativ e charges -q, -2q, -3q,.. ......., .... ..., -12q -12 q fix ed at the position positi on of the corresponding corresp onding numerals on the dial. The clock hands do not disturb the the net field due to point charges. charges. At what what time does the hour hand point in the same direction as electric f ield at the centre of the dial . All the parts of









B.25

The bob of a simple pendulum has a mass of 40 g and a positiv e charge of 4.0 × 10 –6 C. It makes 20 oscillations in 45 s. A verti cal electric f ield pointing upward and of magnitude 2.5 × 104 N/C is switched on. How much time will it now take to compl ete 20 oscillations ?

C.1

Figure represents a square carrying charges +q, +q, –q, –q at its four corners as shown. Then the potential will be zero at points

( A) A, B , C, P an d Q

( B) A , B and C

(C) A, P, C and Q

(D) P, B and Q

C.2

Two Two equal positiv e charges are kept at points A and B. The electric potential at the points between A and B (excluding these points) is studied while moving f rom A to B. The potential (A) conti nuousl y increases (B) continuosl y decreases (C) increases then decreases (D) decreases than i ncreases

C.3

If a uniforml y charged spherical shell of radius 10 cm has a potential V at a point distant 5 cm from its centre, then the potential at a point distant 15 cm from the centre will be : (A)

V 3

(B)

2V 3

(C)

3 2

V

(D) 3V

C.4

A hollow holl ow unif orm ly ch arged arge d sphere sphe re has radi us r. If the t he potenti pote ntial al di ffere ff erence nce between be tween i ts surfa ce and a point at distance 3r from the centre is V, then the electric field in tensity at a distance 3r from the centre is : (A) V/6r (B) V/4r (C) V/3r (D) V/2r

C.5

A hol low sp here o f radiu r adius s 5 cm is uni for ml y charged cha rged such t hat t he poten po ten tial on its i ts surf ace is 10 v olt then potential at centre of sphere will be : (A) Zero (B) 10 v olt (C) Same as at a point 5 cm away from the surface (D) Same as at a point 25 cm away from the centre

C.6

A hollow holl ow sphere is unif orml y charged. charge d. Inside Insid e the sphere (A) The potential is zero but the electric field is finite (B) The electric field is zero but the potential is finite (C) Both the electric field and the potential are finite (D) Both the electric field and the potential are zero

C.7

Two Two spherical conductors of radii 4 m and 5 m are charged to the same potential . If 1 and respective v alues of the surface charge densities on the two conductors, then the ratio

(A) C.8*

5 4

(B)

4 5

(C)

25 16

(D)

 2 are the

1  2 is :

16 25

Whic h of the foll owing quantites do not depend depend on the choice of zero potential or zero potential energy (A) potential at a point (B) potential diff erence between two two points









C.10

The potential diff erence between points A and B in the given uniform ele ctric fiel d is :

( A) Ea C.11

 b2 )

(C) Eb

(D) (Eb / 2 )

A particle of charge Q and mass m travels through a potential dif ference V from rest. The final mome ntum of the particle is : mV

2QV Q m A point charge 20 C is shifted shifted from i nfinity to a point P in an electric field with zero acceleration. If the potential of that point point is 1000 1000 volt, then (i) (i) find out wor work k done done by by external external agent agent again against st electr electric ic field? field? (ii) (ii) what what is the wor work k done done by electr electric ic field? field? (iii) (iii) If the kinetic energy energy of charge particle is found increase by 10 mJ when when it is brought brought from infinity to point P then what what is the total work work done by extenal agent? (iv) iv) what what is the work work done done by electric electric field in the part (iii) (v) If a poin pointt char charge ge 30 C is released at rest at point P, P, then find out its kinetic energy at a large distance? (A)

C.12

( B ) E (a 2

(B) 2Q mV

(C)

2m QV

(D)

C.13

What is the potential at origin i f two equal point charges 'q' are placed at (a, 0) and (–a, 0)?

C.14

Six equal point charges 'q0' are placed at six corners of a regular hexagon of side 'a'. Find out work required to take a point charge 'q' (i) Fro From in infin finity ity to to the the ce centr ntre of hexag xagon. (ii) (ii)

From From infin infinity ity to a poin pointt on on the the axis axis whi which ch is at a dis dista tanc nce e ' 3 a ' from the centre

(iii) iii)

Does Does your your ans answe werr to part part (i) (i) and and (ii) (ii) depen depends ds on on the the path path follow followed ed by by the charge charge..

C.15

12 J of work has to be done against an existing electric f ield to take a c harge of 0.01 C from A to B. How much is the potential difference V B – V A ?

C.16

An electric elect ric fie ld of 30 N/C exists along alo ng the negati ve x-axi x-a xis s in space. Cal culate cula te the potenti al dif fere nce VB – V A where the poin ts A and B are gi ven by, ( a) A = ( 0 , 0 ) ; B = ( 0, 2 m ) (b ) A = ( 4 m , 2m ) ; B = ( 6 m , 5m )

C.17

A particl part icl e of char ge + 3 x 10 -9 C is in a uniform field directed to the the left. It is released released from rest rest and -5 moves a distance of 5cm 5 cm after which its kinetic energy is found to 4.5 x 10 J. (a) What work was was done by the electrical force? (b) What is the magnitude magnitude of the the electrical field? (c) What i s the potential potential of the starting point with respect respect to the end point?

C.18

A positiv posit iv e charge Q = 50  C is located in the xy plane at a point havi ng position vecto r r 0 = 2î 3 jˆ m





where î and jˆ are unit vectors in the positive directions of X and and Y axis respectivel respectivel y. Find: (a) (b)

The electr electric ic inte intens nsity ity vecto vectorr and and its its magnitu magnitude de at a poin pointt of of coord coordina inate tes s (8 (8 m, - 5 m). Work Work don done e by by exter externa nall age agent nt in tran transp spor ortin ting g a char charge ge q = 10 10 C from (8 m, 6 m) to the point











C.20

For an infinite line of charge having charge density  lying along x-axis, the work required in moving charge fr om C to A along ar e CA is :

(A)

q

 0

log e

2

(B)

q 40

log e

2

(C)

q 40

loge 2

(D)

q

1

loge 20 2

C.21

In the electric field of a point charge q a certain charge is carried from point A to B, C, D and E the work done: (q is at the centre of circl e) (A) Is least along the path AB (B) Is least along the path AD (C) Is zero al ong each path AB, AC, AD and AE (D) Is least al ong AE

D.1

If a positive charge is shifted from a low potential region to high potential region. the electrical potential energy : (A) Increases (B) Decreases (C) Rem ains constant (D) May i ncrease or dec rease.

D.2

In bringing an electron towards another electron, electrostatic electrostatic potential energy of the system : (A) Deceases (B) Increases (C) Becom es zero (D) Rem ains same

D.3

A parti cle of m ass 2 g and char ge 1C is held at rest on a frictionl ess horizontal horizontal surface at a distance of 1 m from a fix ed charge 1 mC. If If the particle is released it will be repelled. The speed of the particle when it is at distance of 10 m from the fix ed charge is: ( A) 10 0 m / s (B ) 90 m / s (C) 6 0 m / s (D) 4 5 m / s

D.4

A point ch arge arg e hav ing charge ch arge '–q' '– q' and m ass 'm' is rel eased ease d at rest on the axis of a uniforml y charged fix ring of t otal charge 'Q' and radius 'R' from a distance

3 R . Find out its vel ocity when it reaches to centre

of ring. D.5

A solid uniformly charged non-conduct ing sphere of total charge Q and radius R contains a tunnel of negligi ble diameter. diamet er. If a point charge –q of









E.2

Two Two smooth spherical non conducting shells each of radius R having unif ormly distributed charge Q &  Q on their surfaces are released on a smooth non-conducting surface when the distance between their centres is 10 R. The mass of A is m and that of B is 2 m. Th e speed of A just before A and B collide is: [Neglect gravitational interaction]

2 kQ 2

(A) E.3

(B)

15 mR

4 kQ 2 15 mR

(C)

8 kQ 2

(D)

15 mR

16 kQ2 15 mR

A point c harge har ge q is broug ht from fr om infi i nfi nity nit y and is place pl aced d at the cen tre of a conducti condu cting ng neutra neu trall spher ical shell of i nner radius a and outer radius b, then work done by ex ternal agent is: (A) 0

(C)

(B )

k q2 2b

-

k q2 2a

(D)

k q2 2b k q2 2a

-

k q2 2b

E.4

Two Two positiv e point charges 15 C and 10 C are 30 cm apart. Calculate the work done in bringing them closer to each other by 15 cm.

E.5

Eight equal point charges each of charge 'q' and mass 'm' are placed at eight corners of a cube of side ‘a’. (i) (i) Find Find out out poten potentia tiall ener energy gy of of charge charge syst system em (ii) (ii) Find out out work requir required ed by external external agent agent against against electrost electrostatic atic forces forces and and by electrostatic forces to increase all sides of cube from a to 2a. (iii (iii)) If all the charges charges are releas released ed at rest rest then find out out their their speed when when they they are at the corners of cube of side 2a. (iv) If keeping keeping all other other charges charges fix , charge charge of corner corner 'A' is release released d then find out its speed when it is at infini te distance? (v) If all charges are released released at at rest then then find out their speed speed when when they they are at a very large distance from each other. other.

E.6

You are giv en an arrangment of three point charges q, 2q and x q separated by equal finite distances so that electric potential energy of the system is zero. Calculate the value of x .









G-1

Due to an electric dipole shown in fig., the electric field i ntensity is parallel to dipole axis :

( A) a t P o n l y

( B) at Q o nl y

(C) b ot h at P a nd at Q (D) n ei t he r a t P n or a t Q

G-2

An electric elect ric dipole dip ole consi sts of two opposite char ges each of magni tude 1.0 C separated by a distance of 2.0 cm. The dipole is placed in an ext ernal field of 1. 0 × 105 N/C. The maximum torque on the dipole is : (A) 0.2 × 10 –3 N-m (B) 1.0 × 10 –3 N-m (C) 2.0 × 10 –3 N-m (D) 4.0 × 10 –3 N-m

G-3

A dipol e of electric elect ric dipole dipo le mom ent P is placed in a uniform elect ric field of strength E. If  is the angle between positive directions of P and E, then the potential energy of the e lectric dipole is largest when is : (A) zero (B)  /2 (C)  (D)  /4



G-4

Two opposite opposite and equal charges 4 × 10 –8 coulomb when placed 2 × 10 –2 cm apart form a dipole. If this dipole is placed in an external electric field 4 × 10 8 N/C, the value of maximum torque and the work work done in rotating it through 180º from its initial orientation which is along along electric field will be : (A) 64 × 10 –4 N-m and 44 × 10 –4 J (B) 32 × 10 –4 N-m and 32 × 10 –4 J (C) 64 × 10 –4 N-m and 32 × 10 –4 J (D) 32 × 10 –4 N-m and 64 × 10 –4 J

G-5

The volume ch arge density as a function of distance X from one face inside a unit cube is v arying as shown in the figure. Then the total f lux (in S.I. units) through the cube if 0 = 8.85  10 12 C/m 3 is:

( A) 1 / 4 G-6

( B) 1/ 2

(C) 3/4

(D) 1

Two Two short electric dipoles are pla ced as shown (r is the distance between their c entres). The energy of electric interaction between these dipoles will be: (C is centre of dipole of moment P 2) (A)

2k P1P2 cos 3

r

(B)

2k P1P2 cos r 3









G-9

A charg e ' q ' is carried f rom a poi nt A (r, 135º) to point B (r, 45º) follo wing a path which is a quadrant of 

circle of radius ' r '. r '. If the dipole moment is P, then find out the work done by external agent ?



G-10

Find out force experienced by short dipole P0 in following dif ferent arragnement as shown in figures. 



[Assume point charge is Q, P0 = q0(2a) and P = q(2a)]

(i)

(ii)

(iii)

(iv) 

G-11

Find out the magnitude of electric field intensity at point (2, 0, 0) due to a dipole of dipole mom ent, P = î



3 jˆ kept at origin? Also find out the potential at that point. 

G-12

Four short dipoles each of dipole m oment P are placed at the vertices of a square of side a. The direction of the dipole mom ents are shown in the figure. Find the electric f ield and potential at the centre ‘O’ of the square.

H-1* H-1*

Figure shows shows a charge charge q placed at the centre centre of of a hemisphere hemisphere.. A second second charge charge Q is placed at one of the the positions A, B, C and D. In which position(s) of this second charge, the flux of the electric field through the hemisphere remains unchanged ?









H-3

Select the correct statement : (A) The electric lines of f orce are always closed curves (B) Electric line of force is parallel to equipotential surface (C) Electric line of force is perpendicular to equipotential surface (D) Electric line of force is al ways the the path of a positively charged particle.

H-4

Figure (a) shows an imaginary cube of edge L/2. A unifromly charged rod of l ength L moves towards left at a small but constant constant speed v. At t = 0, the left end just touches the centre of the face of the cube opposite it. Which of the graphs shown shown in fig.(b) represents the flux of t he electric filed through the cube as the rod goes through it ?

(a) (A) a H-5*

( b)

(B) b

(C) c

Figure shows a closed surface surface which intersects a conducting sphere. If a positive charge is placed at the point P, P, the flux of the electic f ield through the closed surface

(A) wil l rem ain zero (C) will becom e neagati v e H-6

(B) wil l become posit iv e (D) wi ll become undef ined

Eight point charges charges (can be assumed as as small spheres spheres uniformly charged and their centres at the corner of the cube) having v alues q each are fixed at v ertices of a cube. The electric flux through square square surface ABCD ABCD of the cube is (A)

H-7

(D) d

q 24 0

(B)

q 12 0

(C)

q 6 0

(D)

q 8 0

A charge Q is place d at a distance of 4R abov e the centr e of a disc of radius ra dius R. The magn itude itu de of flux flu x through the disc is . Now a hemispherical hemispherical shell of radius radius R is placed over the disc such such that it forms a closed surface. The flux through the curved surface taking direction of area vector along outward normal as positive, is











H-11

spherical gaussian surface surface of n which of the following case the flux is maximum through close spherical radius R ?

H-12

What do you predict by the statement about the nature of charge in a close surface. "In a close surface lines which are leaving the surface are double then the lines which are entering in it".

H-13

The electric field in a region is given by E 



E0 x  i . Find the charge contained inside a cubical volum e 

bounded by the surface surface x = 0, x = a, y = 0, y = a, z = 0 and z = a. Take Take E0 = 5 × 103 N/C,  = 2 cm and a = 1 cm.

I .1

Two Two conducting plates X and Y, each having l arge surface area 'A' as shown shown in figure (on one side) are placed parallel to each other. The plate X is given a charge Q whereas the other is neutral. The electric field at a point in between the plates is given by: (A)

(C)

Q

(B)

2 A Q 2 A 0

towards right

(D)

Q 2 A 0 Q 2 0

towards left

towards right

.2

A charge Q is unifroml y distributed distribu ted over ove r a large larg e plastic plasti c plate. The elec tric fi eld at a point P clo se to the the centre of the plate is 10 V/m. If the plastic plate is replaced by a copper plate of the same geometrical geometrical dimensions and carriying the same charge Q, the electric field at the point P will become (A) zero (B ) 5 V / m (C) 10 V/ m (D) 20 V/ m

.3*

A and B are two con centric cent ric spher sp her ical shel ls. A is giv en a charge char ge Q whil e B is uncha rged. rged . If now B is









I.6

.7

I.8

A p ositiv osit iv e point poi nt charge char ge Q is kep t ( as sh own in the fi gure) inside insi de a neutral neut ral conduct cond ucting ing shel l whose centre is at C. An external uniform electric fi eld E is applied. Then (A) force on Q due to E is zero (B) net force on Q i s zero (C) net force acting on Q and conducting shell c onsidered as a system is zero (D) net force acting on the shell due to E is zero. A thin, metall ic spherical shell contains a chrage Q on it. A point charge q is placed at the centre of t he shell and another charge q1 is placed outside it as shown in fig. All t he three charges are positiv e. The force on the charge at the centre is (A) towards lef t (B) towards right (C) upward (D) zero The net charge given to an isolated conduc ting solid sphere: (A) must be distributed uniformly on the surface (B) may be distributed distributed uniformly on the surface (C) must be distributed uniforml y in the volume (D) may be distributed distributed uniformly in the vol ume.

.9

Three identical metal plates with large equal surface areas are kept paralle l to each other as shown in figure. The leftmost plat e is given a charge Q, the rightmost a charge – 2Q and the middle one remains neutral. Find the charge appearing on the outer surface of the rightmost plate.

.10

Figure shows shows two isolated conducting spheres of radius 2cm and 3cm containing charges 10C and 20 C respectiv ely. When the spheres are connected by a conducting wire then find out fol lowing : (i) Ratio of the f inal charge. (ii) Fi nal charge on each sphere. (iii) Ratio of fi nal charge densi ties. (iv) Heat l oss durin g the process.











E J.1*

The electric field intensity at a point in space is equal in magnitude to : (A) The potent ial gradient there (B) T he e l ec t r i c c ha r ge t h er e (C) The force, a unit charge would would experience there (D) The force, an electron would experience there

J.2

The variation of potential with distance r from a fix ed point is shown in Figure. The electric field at r = 5 cm, is : ( A ) ( 2 .5 ) V / c m ( B) ( –2 . 5) V / c m (C) (–2/5) cm (D) (2/5) V/cm

J.3

The electric potential V as a function of distance x (in metre) is given by V = (5x 2 + 10x – 9) volt The value of electric f ield at x = 1 m would be : (A) – 20 v ol t/m (B) 6 v ol t/ m (C) 11 v olt/m (D) –23 v olt/m

J.4

A unifor m electric electr ic field fie ld having hav ing a magnit ude E0 and direction along positive X-axis exists. If the el ectric potential V is zero at x = 0, then its value at x = + x will be : (A) V x = x E0 (B) V x = –xE0 (C) V x = x 2E0 (D) V x = –x 2 E 0

J.5

The electric potential decreases uniformly from 120 V to 80 V as one moves on the X-axis from x = – 1 cm to x = + 1 cm. The electric field at the origin (A) m ust be equa l to 20V/cm (B) m ay be equa l to 20V/cm (C) m ay be greater than 20V/cm (D) m ay be l ess than 20V/cm

J.6

The electric f ield in a region is directed outward and is proportional proportional to the distance r form the origin. Taking the electric potential at the origin to b e zero, the electric potential at a distance r : ( A ) i s u n i f o r m i n t h e r e gi o n (B) i s proporti onal to r 2 (C) is proportional to r (D) increases as one goes away away from the origin.

J.7

If V = x 2y + y 2z then find E (x, y, z)











3.

Three point charges of 1 C, 2 C and 3 C are placed at the the corners of an equilateral triangle of side 1 m. Calculate the work required to move these charges to the corners corners of a smaller equilateral triangle of sides 0.5 m as shown shown in the figure . [ REE '97, 5 ]

4.

Select the correct alternative : (a) A positiv posit iv ely charged char ged thi n met al ri ng of radiu ra dius s R is fi xe d in the xy x yplane with its centre at the origin O . A negatively charged particle P is released from rest at the point (0, 0, z0) where z0 > 0 . Then the motion of P is : (A) periodic, for all val ues of z0 satisfying 0 < z 0  (B) simple harmonic, for all va lues of z0 satisfying 0 < z 0  R (C) approximately simple harmonic, provided z0 << R (D) such that P crosses O & continues to move along the negative z-axis towards x =  

(b)

A charge +q is fi xed at each of th e points x = x0, x = 3x 0, x = 5x 0, ...... ad. inf. on the x-axi s and a charge -q is fixed at each of the points x = 2x 0, x = 4x 0, x = 6x 0, .....ad. inf. Here x 0 is a positive constant. constant. Take Take the electric potential at a point due to a charge Q at a distance distance r from it to be

( A) 0 (c)

Q 4  0 r

. Then the potential at the origin due to the above system of charges is:

( B)

q 8  0 x 0 n 2

(C)



(D)

q n 2 4  0 x 0

A nonconduct noncon duct ing soli d sphere of ra dius R is unif ormly orm ly char ged. The m agni tude of th e elect ric field due to the sphere at a distance r from it s centre : (A) (A) incr increa ease ses s as r incr increa ease ses s, for for r < R (B) (B) decre ecreas ases es as r incr increa eas ses, es, for 0 < r <  (C) decreases as r increases, for R < r < 

(D) is discontinuous at r = R. [ JEE '98Mains, 2+2+2/200 ]









8.

( a)

The dimensions of

(A) M L T 1

(b)

space; E: electric f ield) are:  12   E ( : permittivity of free space; 0

(B) M L2 T  2

(E) M L1 T  2 [ JEE 2000 Scr.,3/105 ] Three charges Q , + q and + q are placed at the vertices of a right  angled isosceles triangle as shown. shown. The net electrostatic energy of the configuration is zero if Q is equal to :

q

(A) 1  2 (c)

2

0

(B)

Four point charges + 8



27 m , 2



3 m, + 2

(C) M L T 2

 2q 2 2

( D)

(C) -2 q

C ,  1  C ,  1 C and 3 m and + 2

M L2 T  1

(D) + q [ JEE 2000 Scr.,3/105 ] + 8 C , are fix ed at the points,

27 m respectively on the y  axis. A particle particle of mass 2

6  10 4 kg and of charge + 0.1 C moves along the -x direction . Its speed at x = +  is v 0 . Find the least value of v 0 for which the particle will cross the origin. Find also the kinetic energy of the particle at the origin. Assume that space space is gravity f ree. Given 1/(4  0) = 9  109 Nm 2/C2 [ JEE 2000 Mains,10/ 100] 9.

( a)

A uniform e lect ric fiel f ield d point ing in posit iv e x-direct x-di rect ion exi sts in a regi on. Let A be the origi n, B be the point on the x-axis at x = + 1 cm and C be the point on the y-axis at y = + 1 cm. Then the potentials at the points A, B & C satisfy : (A) V (B) V (C) V (D) V











11.

Eight charges each of magnitude ‘ q ‘ , are placed at the vert ices of a cube of side a . The nearest neighbours of any charge have opposite sign . Find the work required to dismental t he system . [ JEE 2003Mains, 2/60 ]

12.

A point positiv e charge Q is fixed fixe d at origin and a dipole P is place d at very large l arge distance on x-axis x-axi s with with 

P point away from the origin . Find the kinetic energy of the dipole when it is at a distance ' d ' from origin and at that moment , f ind the force on charge by dipole. 13.

14.

Six charges q,q,q, – q, –q and –q are to be arranged on the vert ices of a regular hexagon PQRSTU such that the electric field at centre is double the field produced when when only charge ‘q’ ‘q’ is placed at vertex R. The sequence of the charges from P to U is ( A) q, – q, q , q , – q, – q (B ) q, q , q , – q , –q , – q (C) –q, q, q , –q, –q, q (D) – q, q, q , q , –q , – q

[ JEE 2003 Mains, 4/60 ]

[JEE 2004 Scr, 3/84]

Three large parallel pl ates have unif orm surface charge densities as shown in the figure. Find out electric fiel d intensity at point P. P.









EC = A.1 C A.5 A A.9 D

A.2 C,D A.6 C

A.3 A A.7 D

A.4 C A.8 D

( 4 î  3 jˆ ) A.10 |F| = 0.18 N, Fˆ  . 5 A.11 Decreased to 50% of initial va lue. A.12 4.95 × 105 N A.13 (i) 3.6 × 10 –6 N (i i ) 2 m / s 2 (iii) No (Magnitude is same but direction is different) A.14 (i) 0,

(ii) (iii)

Kq0 q a2

ED =

 ˆ , 2  0 j

B.20 (i) 2E (ii) For point in side the sphere 2E ; and for outside the sphere B.21

4E 9 B.22 – 25 × 10-9 C 1/ 2

 6 2 m r  0   B.24    e a  

B.23 9:30 B.25 52 s

, a = distance distance of corner corner from centre.

2Kq0 q 2

7 ˆ , 2  0 j

sin 36º,

C.1 B

C.2 D

C.3 B

C.4 A

C.5 B

C.6 B









(iii)

(iv)

E.6



Kq2  3 3  2ma  2

 Kq2  3 1  3 1   3   (v) 3    ma  ma  2 3 2 3

2Kq2

2

F.3 W el =

.12 I.12

( - x) A, x) A, x A, x A, - xA,(x - 2 ) A where x =(2 =( 2  0 E+3)/2

.13 I.13

1500 V, 1200 V

J.1 AC

J.2 A

J.5 BC

J.6 C

I.14

J.3 A

V=

1

q

4   0 r

J.4 B

J.7 –2xy –2x y î – (x 2 + 2yz) jˆ – y 2 kˆ  J.8 (i) – 4( î – 2 kˆ ) (ii ) E  8r ˆ

3

F.1 B

F.2 D

J.9 –100 V

q( q0  q / 2)  q / 2) , W ext =– 80R 8 0R

q(q0

F.4 K.E. = G-1 C G-4 D

   3 1

1

Q2

2 40 d G-2 C G-5 C

q2 F.5 160R G-3 C G-6 B

G-7 ABD

J.10

 2r 3 3

1.

D

2.

A

3.

4.

(i) A, C

(ii) D

(iii) A, C

5.

( a ) U2 =

a2 Q 2 8  0 R

+C

 1 a n     1 a 

J.11 – 2xy + C

9.9 x 1010 J 2

where a =

R r R









Electrostatics Exercise

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