# Chapter 3 Electric Charges and Fields (Electrostatics Part 3) – Physics free study material by TEACHING CARE online tuition and coaching classes

Chapter 3 Electric Charges and Fields (Electrostatics Part 3) – Physics free study material by TEACHING CARE online tuition and coaching classes

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Example: 63     Two point charges +4q and +q are placed at a distance L apart. A third charge Q is so placed that all the three charges are in equilibrium. Then location and magnitude of third charge will be

• At a distance Lfrom +4q charge, 4q

3                             9

• At a distance

L from +4q charge, – 4q

3                                 9

• At a distance

2L from +4q charge, – 4q

• At a distance

3

2L from +q charge,

3

9

• 4q 9

Solution: (c)      Let third charge be placed at a distance

x1 from +4q charge as shown

Now x

=         L       2L

Þ x   = L                                                                             Q                                                         +q

1                             3          2      3

1 +

+4q

x1                          x2

L

æ L / 3 ö 2     4q                     4q

For equilibrium of q, Q = +4qç               ÷

è  L   ø

=  9  Þ Q = –  9 .

Example: 64     A drop of

10-6 kg water carries

10-6 C charge. What electric field should be applied to balance it’s

weight (assume g = 10 m/sec2)

(a)

10 V / m, Upward               (b)

10 V / m, Downward       (c) 0.1 V/m Downward (d)

0.1V / m,

Upward

Solution: (a)      In equilibrium QE = mg

E mg

Q

= 10-6 ´ 10 10-6

= 10 V/m; Since charge is positive so electric field will be upward.

Example: 65     A charged water drop of radii 0.1 mm is under equilibrium in some electric field. The charge on the drop

is equivalent to electronic charge. The intensity of electric field is                                                 [RPET  1997]

(a)

1.61 N / C

(b)

25.2 N / C

æ 4 pr 3 r ö.g

(c)

262 N / C

(d)

1610 N / C

mg      ç 3          ÷         4 ´ (3.14)(0.1 ´ 10 6 )3 ´ 103 ´ 10

Solution: (c)       In equilibrium QE = mg ;      E =

è             ø     =                                                               = 262 N/C

Q               Q              3

1.6 ´ 10 19

Example: 66     The bob of a pendulum of mass 8 mg

carries an electric charge of

39.2 ´ 10 -10

coulomb in an electric

Solution: (b)

field of  20 ´ 103 volt / meter and it is at rest. The angle made by the pendulum with the vertical will be (a) 27o        (b) 45o  (c) 87o                                      (d) 127o

T sinq = qE , T cosq = mg

qE

\     tanq =

mg

qE

mg

tanq = 39.2 ´ 10-10 ´ 20 ´ 103 = 1

8 ´ 10-6 ´ 9.8

Þ  q = 45o

Example: 67     Two small spherical balls each carrying a charge Q = 10 mC (10 micro-coulomb) are suspended by two

insulating threads of equal lengths 1 m each, from a point fixed in the ceiling. It is found that in equilibrium threads are separated by an angle 60o between them, as shown in the figure. What is the

1

tension in the threads. (Given : (

) = 9 ´ 109 Nm / C 2 )

• 18 N
• 8 N

(c) 0.18 N

(d) None of these

Solution: (b)      From the geometry of figure

r = 1m

4pe 0

In the condition of equilibrium T sin 30o = Fe

0o

T ´ 1

2

= 9 ´ 109.

(10 ´ 10 -6 )2

12

+10 mC

Þ T= 1.8 N

Example: 68 Two similar balloons filled with helium gas are tied to L m long strings. A body of mass m is tied to another ends of the strings. The balloons float on air at distance r. If the amount of charge on the balloons is same then the magnitude of charge on each balloon will be

é mgr 2

ù1 / 2

(a)

ê 2k

 ê
 ë

é 2k

tanq ú

úû

ù1 / 2

(b) ê mgr 2 tanq ú

(c)

ë

é mgr

êë 2k

é 2k

û

ù1 / 2

cotq úû

ù1 / 2

(d) ê mgr tanq ú

ë                  û

Solution: (a)      In equilibrium                                                                                          R                                                     R

r

2R = mg

…. (i)

Fe = T sinq …. (ii)

R = T cosq

…. (iii)        F

T sin

From equation (i) and (iii)

T cos

T

L  q  q   L

2T cosq = mg

…. (iv)

m

Dividing equation (ii) by equation (iv)                                                                        m

Q2

• F 1            k   2

æ mgr 2

ö1 / 2

 è
 ø

tanq =  e   Þ         tanq =  r      Þ q = ç              tanq ÷

• mg 2             mg

ç   2k              ÷

# Time Period of Oscillation of a Charged Body.

#### (1)    Simple pendulum based : If a simple pendulum having length l and mass of bob m oscillates about it’s mean position than it’s time period of oscillation T = 2π

 Case – 1 : If some charge say +Q is given to bob and an electric field E is applied in the direction as shown in figure then equilibrium position of charged bob (point charge) changes from O to O¢.     E d q   l                           O¢                      QE mg¢ O                               mg   On displacing the bob from it’s equilibrium position 0¢. It will oscillate under the effective acceleration g¢, where mg’ = (mg)2 + (QE)2   Þ g’ = g 2 + (QE / m)2   Hence the new time period is T1 = 2p l g’   T1 = 2π                  l               (g 2 + (QE/m)2 )2 1     Since g’ >g, hence T1 < T i.e. time period of pendulum will decrease. Case – 2 : If electric field is applied in the downward direction then. Effective acceleration  g’ = g + QE / m                                        q  l       T So new time period                                               E   T2 = 2π              l          g + (QE/m) mg + QE T2 < T Case – 3 : In case 2 if electric field is applied in upward direction then, effective acceleration.  g’ = g – QE / m So new time period                         q                     QE l T3 = 2π              l                                                              E   g – (QE/m) T3 > T mg Case – 4 : In the case 3, QE if T3 = T  i.e., 2p             l                   O¢¢ 2                   (g – QE / m) 1         l                                                                                  E   = 2 2p   g    Þ QE = 3 mg                                mg i.e., effective vertical force (gravity + electric) on the bob = mg – 3 mg = – 2 mg, hence the equilibrium position O¢¢ of the bob will be above the point of suspension and bob will oscillate under on effective acceleration 2g directed upward. Hence new time period T4 = 2π   l  ,  T4 < T 2g

• Charged circular ring : A thin stationary ring of radius R has a positive charge +Q If a negative charge – q (mass m) is placed at a small distance x from the centre. Then motion of the particle will be simple harmonic motion.

E =    1

4pe 0

.

(x 2

Qx

 )

3

+ R 2 2

#### Since x<< R, So

x 2 neglected hence

E =    1

4pe 0

. Qx R 3

1

4pe 0

. Qx R 3

#### Þ F µ –x hence motion is simple harmonic

Having time period T = 2π

#### (3)    Spring mass system : A block of mass m containing a negative charge – Q is placed on a frictionless horizontal table and is connected to a wall through an unstretched spring of spring

constant k as shown. If electric field E applied as shown in figure the block experiences an electric force, hence spring compress and block comes in new position. This is called the equilibrium position of block under the influence of electric field. If block compressed further or stretched, it execute oscillation having time

period T = 2π

# Neutral Point.

#### . Maximum compression in the spring due to electric field = QE

k

A neutral point is a point where resultant electrical field is zero. It is obtained where two electrical field are equal and opposite. Thus neutral points can be obtained only at those points where the resultant field is subtractive. Thus it can be obtained.

# Zero Potential Due to a System of Two Point Charge.

#### If both charges are like then resultant potential is not zero at any finite point because potentials due to like charges will have same sign and can therefore never add up to zero. Such a point can be therefore obtained only at infinity.

If the charges are unequal and unlike then all such points where resultant potential is zero lies on a closed curve, but we are interested only in those points where potential is zero along the line joining the two charges.

#### Two such points exist, one lies inside and one lies outside the charges on the line joining the charges. Both the above points lie nearer the smaller charge, as potential created by the charge larger in magnitude will become equal to the potential created by smaller charge at the desired point at larger distance from it.

N

Example: 69 Two similar charges of +Q as shown in figure are placed at points A and B. – q charge is placed at point C midway between A and B. – q charge will oscillate if

• It is moved towards A
• It is moved towards B
• It is moved along CD
• Distance between A and B is reduced

Solution: (c)      When – q charge displaced along CD, a restoring force act on it which causes oscillation.

Example: 70     Two point charges (+Q) and (– 2Q) are fixed on the X-axis at positions a and 2a from origin respectively. At what position on the axis, the resultant electric field is zero

• Only x = 2 a
• Only

x = – 2 a

• Both

x = ± 2 a

x = 3a

2

only

Solution: (b)      Let the electric field is zero at a point P distance d from the charge +Q so at P.

k.Q + k(2Q) = 0

Þ     1 =       2

Þ d =        a

d 2     (a + d)2

d 2     (a + d)2

(     – 1)

Since d > a i.e. point P must lies on negative x-axis as shown at a distance x from origin hence

x = d a =

a      a =

2a . Actually P lies on negative x-axis so x = –

2 a .

Example: 71     Two charges 9e and 3e are placed at a distance r. The distance of the point where the electric field intensity will be zero is                                                                                                         [MP PMT 1989]

(a)

(c)

from 9e charge                                          (b)

from 3e charge                                          (d)

r           from 9e charge

r           from 3e charge

Solution: (b)      Suppose neutral point is obtained at a distance x1 from charge 9e and x 2 from charge 3e

By using

x                    r                    r

 =
 =

x1                      =

æ         1 ö

1 +                  1 +

ç1 +         ÷                                                               r

è          3 ø

Example: 72     Two point charges – Q and 2Q are separated by a distance R, neutral point will be obtained at

• A distance of

• A distance of

• A distance of

R        from – Q charge and lies between the charges.

R        from – Q charge on the left side of it

R       from 2Q charge on the right side of it

• A point on the line which passes perpendicularly through the centre of the line joining – Q and 2Q

Solution: (b)      As already we discussed neutral point will be obtained on the side of charge which is smaller in magnitude i.e. it will obtained on the left side of – Q charge and at a distance.

l =         R

– 1

Þ l =        R

Example: 73     A charge of + 4mC is kept at a distance of 50 cm from a charge of – 6mC. Find the two points where the potential is zero

• Internal point lies at a distance of 20 cm from 4mC charge and external point lies at a distance of 100

cm from 4mC charge.

• Internal point lies at a distance of 30 cm from 4mC charge and external point lies at a distance of 100

cm from 4mC charge

• Potential is zero only at 20 cm from 4mC charge between the two charges
• Potential is zero only at 20 cm from – 6mC charge between the two charges

Solution: (a)      For     internal     point    X,

x1 =         x        =   50   = 20cm

and     for     external     point     Y,

 ç

æ Q2 Q

ö

+ 1÷

6 + 1

4

x   =         x        =   50   = 100cm

è   1        ø

1     æ Q          ö       6

ç   2 – 1÷

4 – 1

è Q1         ø

100cm

50cm

Tricky example: 9

Two equal negative charges – q are fixed at points (0, a) and (0, – a) on the y-axis. A positive charge Q

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

[IIT-JEE 1984, Bihar MEE 1995, MP PMT 1996]

• Execute simple harmonic motion about the origin
• Move to the origin and remains at rest
• Move to infinity
• Execute oscillatory but not simple harmonic

Solution: (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.

F Þ 2F cosq = 2   1        qQ      ´             x

A q

i.e., F = –

4pe 0 (a 2 + x 2 )

1    .      2qQx

(a 2 + x 2 )1 2

a

4pe 0

(a 2 + x 2 )3   2

O                      q       Q

x                                        C

As the restoring force F is not linear, motion will be oscillatory (with             a                                  2a

amplitude 2a) but not simple harmonic.

q

B

# Electric Potential Energy.

#### (1)    Potential energy of a charge : Work done in bringing the given charge from infinity to a point in the electric field is known as potential energy of the charge. Potential can also be written as potential energy

per unit charge. i.e.

V = W = U .

Q      Q

• Potential energy of a system of two charges : Since work done in bringing charge Q2 from ¥ to

point B is W = Q V , where V

i.e. V

#### =    1    Q1

So,

2  B                                B

W = U2 =

 0

#### 1

4pe0

Q1Q2

r

1             B          4pe     r

will be U1 =

1

4pe0

Q1Q2

r

#### Hence potential energy of Q1 = Potential energy of Q2 = potential energy of system

U = Q1Q2 )

r

U = k Q1Q2 (in C.G.S.

r

#### Note : @Electric potential energy is a scalar quantity so in the above formula take sign of Q1 and Q2.

• Potential energy of a system of n charges : In a system of n charges electric potential energy is calculated for     each  pair    and then         all   energies   so      obtained                       are              added   i.e.

U    1   é Q1Q2  + Q2Q3  +……….. ù

#### and in case of continuous distribution of charge. As

dU = dQ.V        Þ

 ë
 û

4pe0 ê

U = ò V dQ

r12

r23                     ú

#### e.g. Electric potential energy for a system of three charges

 ë
 û

Potential energy = 1 é Q1Q2 + Q2Q3 + Q3Q1 ù

4pe0 ê

r12

r23

r31   ú

#### While potential energy of any of the charge say Q

is  1    é Q1Q2 + Q3Q1 ù

 0 ë
 û

1     4pe ê

r12

r31   ú

n(n – 1)

#### 2

number of

• Electron volt (eV) : It is the smallest practical unit of energy used in atomic and nuclear As

#### electron volt is defined as “the energy acquired by a particle having one quantum of charge 1e when

accelerated by 1volti.e. 1eV = 1.6 ´10-19 C ´ 1J

C

#### = 1.6 ´ 10-19J

= 1.6 ´ 10–12 erg

volt)

Energy acquired by a charged particle in eV when it is accelerated by V volt is E = (charge in quanta) × (p.d. in

 S.No. Charge Accelerated by p.d. Gain in K.E. (i) Proton 5 ´ 104 V K = e ´ 5 ´ 104 V = 5 ´ 104 eV = 8 ´ 10–15 J [JIPMER 1999] (ii) Electron 100 V K = e ´ 100 V = 100 eV = 1.6 ´ 10–17 J [MP PMT 2000; AFMC 1999] (iii) Proton 1 V K = e ´ 1 V = 1 eV = 1.6 ´ 10–19 J [CBSE 1999] (iv) 0.5 C 2000 V K = 0.5 ´ 2000 = 1000 J [JIPMER 2002] (v) a-particle 106 V K = (2e) ´ 106 V = 2 MeV [MP PET/PMT 1998]

• Electric potential energy of a uniformly charged sphere : Consider a uniformly charged sphere of radius R having a total charge Q. The electric potential energy of this sphere is equal to the work done in bringing the charges from infinity to assemble the

U =    3Q2

20pe0R

### (6) Electric potential energy of a uniformly charged thin spherical shell :

U =     Q2

8pe0 R

Ue  =

U

= 1 e

2 0

= 1 e

#### 2

0e r E 2

Q                                                                     q

U = 0

Example: 74     If the distance of separation between two charges is increased, the electrical potential energy of the system                                                                                                                                   [AMU 1998]

(a) May increases or decrease                                           (b) Decreases

(c) Increase                                                               (d) Remain the same

Solution: (a)      Since we know potential energy

= k. Q1Q2

r

As r increases, U decreases in magnitude. However depending upon the fact whether both charges are similar or disimilar, U may increase or decrease.

Example: 75     Three particles, each having a charge of 10mC are placed at the corners of an equilateral triangle of side

10cm. The electrostatic potential energy of the system is (Given  1   = 9 ´ 109 N m2 /C 2 )

4pe 0

(a) Zero                              (b) Infinite                        (c) 27 J                                (d) 100 J

[AMU 1998]

Solution: (c)      Potential energy of the system,

U = 9 ´ 109 é(10 ´ 10 -6 )2 ´ 3ù

= 27 J

10mC

 ê
 ú
 ë
 û

ê         0.1            ú

10mC

10 cm

10 cm

10 cm

10mC

Example: 76     Three charges Q, +q and +q are placed at the vertices of a right-angled isosceles triangle as shown. The

net electrostatic energy of the configuration is zero if Q is equal to                                         [IIT (Screening) 2000]

• q

• – 2 q

(d) +q

Solution: (b)      Potential energy of the configuration U = k. Qq + k.q 2 + k. Qq = 0 Þ Q =

a          a            a

Example: 77     A charge 10 e.s.u. is placed at a distance of 2 cm from a charge 40 e.s.u. and 4 cm from another charge of 20 e.s.u. The potential energy of the charge 10 e.s.u. is (in ergs)

(a) 87.5                              (b) 112.5                         (c) 150                         (d) 250

Solution: (d)     Potential energy of 10 e.s.u. charge is

U = 10 ´ 40 + 10 ´ 20 = 250 erg.

10 esu

2            4

20 esu

Example: 78     In figure are shown charges q1 = + 2 × 10–8 C and q2 = – 0.4 × 10–8 C. A charge q3 = 0.2 × 10–8 C in moved along the arc of a circle from C to D. The potential energy of q3                                                           [CPMT 1986]

• Will increase approximately by 76%
• Will decreases approximately by 76%
• Will remain same
• Will increases approximately by 12%

Solution: (b)      Initial potential energy of q       U   = æ q1q3 + q2 q3 ö ´ 9 ´ 109                  q

 ø

3          i         ç

è

0.8        1    ÷                                         3

æ q q        q  q   ö                                          C

Final potential energy of q3 U f = ç 1 3 + 2 3 ÷ ´ 9 ´ 109

è

Change in potential energy = Uf Ui

0.8

0.2 ø

Now percentage change in potential energy = U f  Ui ´ 100

ui

A                                   B

D

q1                60 cm             q2

80 cm

q q æ 1 – 1ö ´ 100

 è
 ø

2 3 ç 0.2       ÷

=

On putting the values ~– – 76%

 3 ç 0.8

q  æ q1

è

q2 ö

 1 ÷

ø

# Motion of Charged Particle in an Electric Field.

### (1) When charged particle initially at rest is placed in the uniform field :

#### Let a charge particle of mass m and charge Q be initially at rest in an electric field of strength E

Fig. (A)                                                                 Fig. (B)

• Force and acceleration : The force experienced by the charged particle is F = QE . Positive charge

#### experiences force in the direction of electric field while negative charge experiences force in the direction opposite to the field. [Fig. (A)]

Acceleration produced by this force is a = F

m

QE

m

#### Since the field E in constant the acceleration is constant, thus motion of the particle is uniformly accelerated.

• Velocity : Suppose at point A particle is at rest and in time t, it reaches the point B [Fig. (B)]

#### V = Potential difference between A and B; S = Separation between A and B

• By using v = u + at ,

v = 0 + Q E t ,               Þ

m

v QEt

m

• By using v2 = u2 + 2as , v2 = 0 + 2 ´ QE´ s

m

v2 = 2QV

m

V

 s ý
 E =
 þ
 Þ
 ì             ü

íQ                    v =

#### î

• Momentum : Momentum p = mv,

p = m ´ QEt = QEt m

or p = m ´            =

• Kinetic energy    :      Kinetic     energy    gained    by     the     particle     in     time    t       is

K 1 mv 2 = 1 m (QEt)2

Q 2 E 2 t 2          or

K 1 m ´ 2QV

= QV

2            2         m                2m                                                     2          m

### (2) When a charged particle enters with an initial velocity at right angle to the uniform field :

#### When charged particle enters perpendicularly in an electric field, it describe a parabolic path as shown

• Equation of trajectory : Throughout the motion particle has uniform velocity along x-axis and horizontal displacement (x) is given by the equation x = ut

#### Since the motion of the particle is accelerated along y–axis, we will use equation of motion for uniform

acceleration to determine displacement y. From S = ut + 1 at 2

We have u = 0

y = 1 at 2

#### 2

i.e., displacement along y-axis will increase rapidly with time (since y µ t 2 )

x

1 æ QE ö æ x ö 2

2 è m ø è u ø

#### (ii)   Velocity at any instant : At any instant t, vx

= u and vy

QEt

m

So            v =| v|=                 =

If b is the angle made by v with x-axis than tan b = vy = QEt .

vx             mu

r

Example: 79     An electron (mass =

9.1 ´ 10 -31 kg

and charge =

1.6 ´ 10 19 coul.)

is sent in an electric field of

intensity 1 ´ 106 V / m. How long would it take for the electron, starting from rest, to attain one–tenth the velocity of light

(a)

1.7 ´ 10 12 sec

(b)

1.7 ´ 10 6 sec

1.7 ´ 10 8 sec

1.7 ´ 10 10 sec

Solution: (b)      By using v = QEt Þ 1 ´ 3 ´ 108 = (1.6 ´ 10 -19 ) ´ 106 ´ t Þ t = 1.7 ´ 10 10 sec.

m           10

9.1 ´ 10 -31

Example: 80     Two protons are placed

10 10 m

apart. If they are repelled, what will be the kinetic energy of each

proton at very large distance

(a)

23 ´ 10 -19 J

(b)

11.5 ´ 10 -19 J

(c)

2.56 ´ 10 -19 J

(d)

2.56 ´ 10 -28 J

Solution: (d)     Potential energy of the system when protons are separated by a distance of 10 -10 m is

9                     -19 2

U 9 ´ 10 ´ (1.6 ´ 10      )   = 23 ´ 10 -19 J

10 -10

P+                                                                           p+

According to law of conservation of energy at very larger distance, this energy is equally distributed in both the protons as their kinetic energy hence K.E. of each proton will be 11.5 ´ 10 -19 J.

Example: 81     A particle A has a charge +q and particle B has charge +4q with each of them having the same mass m. When allowed to fall from rest through the same electrical potential difference, the ratio of their speeds

v A     will becomes                                                                                                     [BHU 1995; MNR 1991]

vB

(a) 2 : 1                              (b) 1 : 2                           (c) 1 : 4                        (d) 4 : 1

Solution: (b)      We know that kinetic energy

K = 1 mv2 = QV .   Since,   m   and V   are   same   so,

2

v2 µ Q Þ

vA  =

vB

=            = 1 .

2

Example: 82     How much kinetic energy will be gained by an a – particle in going from a point at 70 V to another point

at 50 V                                                                                                                                                                            [RPET 1996]

• 40 eV (b) 40 keV                               (c) 40 MeV                          (d) 0 eV

Solution: (a)      Kinetic energy K = QDV

Þ K = (2e)(70 – 50) V

= 40 eV

Example: 83     A particle of mass 2g and charge 1m C is held at a distance of 1 metre from a fixed charge of 1mC. If the particle is released it will be repelled. The speed of the particle when it is at a distance of 10 metres from

the fixed charge is                                                                                                                   [CPMT 1989]

(a) 100 m/s                                  (b) 90 m/s                                (c) 60 m/s                            (d) 45 m/s

Solution: (b)      According to conservation of energy

1 mC             1 mC

A

Fixed charge            1m

Moving charge                         B

10m

Energy of moving charge at A = Energy of moving charge at B

9 ´ 109 ´ 10-3 ´ 10-6 = 9 ´ 109 ´ 10-3 ´ 10-6 + 1 ´(2 ´ 103) v2

1                              10          2

Þ      v2 = 8100     Þ v = 90 m/sec

# Force on a Charged Conductor.

#### To find force on a charged conductor (due to repulsion of like charges) imagine a small part XY to be cut and just separated from the rest of the conductor MLN. The field in the cavity due to the rest of the conductor is E2, while field due to small part is E1. Then

Inside the conductor E = E1 – E2 = 0 or

s

E1 = E2

 e

Outside the conductor E = E1 + E2 =

0

## s

 0

Thus E1 = E2 = 2e

s dA

#### placed in the cavity MN having field E2).

Thus force dF = (s dA)E

or dF = s 2

#### dA . The force per unit area or electric pressure is

dF s 2

 0

2               2e                                                                                dA       2e

0

#### The force is always outwards as (±s )2 will try to expand the charged body.

is positive i.e., whether charged positively or negatively, this force

A soap bubble or rubber balloon expands on given charge to it (charge of any kind + or –).

# Equilibrium of Charged Soap Bubble.

#### For a charged soap bubble of radius R and surface tension T and charge density s . The pressure due to

surface tension 4 T

R

#### and atmospheric pressure

Pout act radially inwards and the electrical pressure

(Pel )

4T – s 2

in        out

R       2e0

4T – s 2

in       out

excess

R       2e0

P   = P

i.e., P

4Ts 2

in        out

excess

R        2e0

#### This result give us the following formulae

8 e0 T

• Radius of bubble R = s 2

• Surface tension T = s2 R

8e0

#### (3)    Total charge on the bubble Q = 8pR

• Electric field intensity at the surface of the bubble E = =