Example 25.1: A proton, of mass 1.67×10-27 Kg, enters the region between two parallel plates a distance 20 cm apart. There is a uniform electric field of 3× 105 V/m between the plate, as shown in Fig25.6 If the initial speed of the proton is 5×106 m/s, what is its final speed?

smv

sm

Kg

VCsm

m

Vqv

VEdV

iVqmvmv

VqVqU

UK

f

i

if

/106

/105.36

1067.1

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2 vhave we(i) From

106

)( 2

1

2

1

)0 ,: ( ,0

0

6

2212

27

41926

22

f

4

22

Example 25.2:Three point charges q1=1μC, q2=-2μC, and q3=3μC are fixed at the positions shown in Fig. 25.13a (a) What is the potential at point p at the corner of the rectangle? (b) How much work would be needed bring a charge q4=2.5μC from infinite and to place it at p? (c) What is the total potential energy of q1,q2 and q3?

particles) theseparate torequired work (external

J10-1.41U

J10-13.5 Uand 104.5

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0.19JV)10C)(7.6510(2.5VqW

0,V , case In the ).V-q(V(b)W

V107.65V is potential totalThe

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)10)(/100.9()(

2-

3-

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13

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12

23

32

13

31

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3

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3

3

3

2

3

6229

1321

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Jm

mNU

r

qkq

r

qkq

r

qkqU

andVV

Vm

CCmNVVVVVa p

EXAMPLE 25.3: In 1913, Niels Bohr proposed a model of the hydrogen atom in which an electron orbit a stationary proton in a circular path. Find the total mechanical energy of the electron given that the radius of the orbit is 0.53×10-10m

eVJm

CCmN

ther

KemvK

r

mv

r

ke

ir

keU

6.131018.21006.1

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ke-UK

isenergy mechanical total22

1

)(

18

10

2192292

22

2

2

2

2

Example 25.4:The potential due to a point charge is given by V=kQ/r. Find:(a) the radial component of the electric field; (b) the x component of the electric field.

3231222

21222

21222

2

,

)(

r

kQx

)zy(x

kQ x

x

VE

)zy(x

kQV

kQ/rV)zy(x r

(b)r

kQ

dr

dVE

a

/x

/

/

r

Example 25.5: A nonconducting disk of radius a has uniform surface charge density σC/m2. What is the potential at a point on the axis of the disk at a distance y from its center?

2

2

22/1

2

22/122

n

1/222

01/222

02/122

2/122

2/122

aQ where

....)2

1()1()(

1z)(1 a,yfor

y-)y(ak2

)y(xk2 )(

2

)(

)2(

)2(,)(

y

kQV

y

ay

y

ayya

zsmallfornz

yx

xdxkV

yx

xdxk

r

kdqdV

xdxdAdqyxr

aa

Example 25.6:A shell of radius R has a charge Q uniformly distributed over its surface. Find the potential at a distance r>R from its center.

Q chargepoint a todue potential theaseffect same the

(25.17) R)(r

1

r

kQ-)V(V(r)

have we0,)V( Since .

ˆ

r

2

2

r

kQV

rkQdr

drEsdE

rr

kQE

r

r

Example 25.7: A metal sphere of radius R has a charge Q. Find its potential energy.

QVR

kQdq

R

kqW

dqRkqVdqdW

RkqV

Q

2

1

2

)/(

/

2

0

DISCUSSION

• RQ ELECTRIC POTENTIAL I,II

Exercises of chapter 25

• Questions: 7,9,15

• Exercises:5,11,35,46,48,50,53,58

• Problems:5,6,7,9,11,12,16