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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?
K  U  0
U  qV  0, ( q : , V  0)
1 2 1 2
mv f  mvi   qV
(i )
2
2
V   Ed  6  10 4V
2qV
From (i) we have vf2  vi2 
m
2(1.69  10 19 C )( 6  10 4V )
6
2
 (5  10 m / s ) 
1.67  10 27 Kg
 36.5  1012 m 2 / s 2
v f  6  10 6 m / s
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?
(9.0  109 N  m 2 / C 2 )(10 6 C )
(a )V p  V1  V2  V3  V1 
4m
 2.25  103V
V2  3.6  103V , and V3  9  103 V.
The total potential is Vp  7.65  103 V
(b)Wext  q(Vf - Vi ). In the case , Vi  0,
Wext  q 4 Vp  (2.5  10 -6 C)(7.65  103 V)  0.19J
(c)U  U12  U13  U 23 
kq1q2 kq1q3 kq2 q3


r23
r13
r12
(9.0  109 N  m2)( 2  10 6 )
 6  10 3 J
U12 
3m
U13  5.4  10 3 J and U 23  -13.5  10 -3 J
U  -1.41  10 -2 J
(external work required to separate the particles)
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
ke 2
U 
(i )
r
ke 2 mv2

2
r
r
1 2 Ke2
K  mv 
2
2r
the total mechanical energy is
ke 2
(8.99  109 N  m 2 / C 2 )(1.60  10 19 C ) 2
KU
2r
1.06  10 10 m
 2.18  10 18 J  13.6eV
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.
(a)
dV
Er  
dr
(b)
kQ
 2
r
r  (x 2  y 2  z 2 )1 / 2 ,
V  kQ/r
kQ
V 2
(x  y 2  z 2 )1 / 2
V
kQ x
kQx
Ex  
 2
 3
2
2 31 / 2
x
(x  y  z )
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?
r  ( x 2  y 2 )1/ 2 , dq  dA   (2xdx)
kdq k (2xdx)
dV 
 2
r
( x  y 2 )1/ 2
a
xdx
2
2 1/2 a

V  2k  2

2

k

(x

y
) 0
2 1/ 2
0 (x  y )
 2k (a 2  y 2 )1/2 - y
for y  a, (1  z) n  1  nz for small z
(a  y )
2
V
2 1/ 2
a 2 1/ 2
a2
 y (1  2 )  y (1  2  ....)
y
2y
kQ
where Q  a 2
y
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.
 kQ
E  2 rˆ
r
 
E  ds  Er dr. Since V( )  0, we have
r
kQ
 1
V(r)  V( )  - 2 dr  kQ 
r
 r 

r
V
kQ
r
(r  R)
(25.17)
the same effect as the potential due to a point charge Q
Example 25.7: A metal sphere of radius R has a charge Q. Find its potential
energy.
V  kq / R
dW  Vdq  (kq / R)dq
W  0
Q
kq
kQ 2 1
dq 
 QV
R
2R 2
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