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PHYS 196 Home Work 5A
1. A semi-circular arc carries a uniform linear charge distribution of 5.0nC / m . Find the potential at the
center.
2. A charge of  10.0C is uniformly distributed on a thin spherical shell of radius 12.0cm. (a) What is the
magnitude of the electric field just outside and just inside the shell? (b) What is the electric potential just
outside and just inside the shell? (c) What is the electric potential and electric field at the center of the
shell? (d) What is the electric potential and electric field at a distance 24cm from the center of the shell?
3. What is the maximum potential that the dome of a van de Graaf generator can be safely charged to if its
radius is 20cm and the breakdown electric field in air is 3.0MV/m?
4. Two spherical conductors are widely separated. One has radius R and the other 2 R . Initially, the small
spheres carries a charge Q while the larger sphere is neutral. They are then connected by a long metallic
wire. What is now the charge on the small sphere?
5. A spherical conductor of radius R1 is charged to 20 kV. When it is connected by a long, very thin
conducting wire to an uncharged second conducting sphere far away, its potential drops to 12 kV. What
is the radius of the second sphere?
6. A solid sphere of radius R carries a total charge Q uniformly distributed throughout its volume. Find the
potential at a point a distance R 2 from the center.
7. A configuration of conductors consists of a thin spherical shell of radius a , carrying a uniformly
distributed total charge Q surrounded by a concentric thick spherical shell with inner radius b and outer
radius c . Find the electric field and electric potential at a point a distance r from the center separately
for the four regions r  a, a  r  b, b  r  c, c  r when the total charge on the outer shell is (a)
zero, (b)  Q , and (c) Q .
8. How much work is required to put the two protons in place (against their mutual electric repulsion)
inside an alpha particle, assuming the distance between them is 1.0  1013 cm . Express your answer in
eV.
9. Three point charges  2.0C , 3.0C and 4.0 C lie on the vertices of an equilateral triangle of side
2.0m. How much work has been done to establish this configuration?
10. Charge is supplied to the metal dome of a Van de Graaf generator by the belt at the rate of 200μC/s
when the potential difference between the belt and the dome is 1.25MV. The dome transfers charge to
the atmosphere at the same rate, so the 1.25MV potential difference is maintained. What minimum
power is needed to drive the moving belt and maintain the 1.25 MV potential difference?
11. (Difficult)Show that the total work needed to assemble a uniformly charged sphere of total charge Q and
radius R is given by 3Q 2 200 R 
Answers
1. 140V
2. 6.25kV / m, 0kV / m, 750V , 750V , 1.56kV / m, 375V
3. 600kV
4. Q 3
5. R2  2R1 3
6. 11kQ 8R
7. (a) 0, kQ r 2 , 0, kQ r 2 , kQ(1 a  1 c  1 b), kQ(1 r  1 c  1 b), kQ c, kQ r
(b) 0, kQ r 2 , 0, 0, kQ(1 a  1 b) , kQ(1 r  1 b) , 0,0
(c) 0, kQ r 2 , 0, 2kQ r 2 , kQ(1 a  2 c  1 b) , kQ(1 r  2 c  1 b), 2kQ c, 2kQ r
8. 1.44MeV
9.  9mJ
10. 250W
1
PHYS 196 Home Work 5B
1. A thin rod carrying electric charge lies on the x-axis, with one end at the origin, and another at the point
x = a . The linear charge density on the rod is given by l ( x ) = C x 2 + ax . Find (a) the total charge q in
(
)
terms of C , and (b) the electric potential at the following points: the origin, the point x = 2a , the point
x = -a . Express your answers in (b) in terms of q .
2. A thin rod carrying electric charge lies on the x-axis, with one end at the origin, and another at the point
2
x = 2a . The linear charge density on the rod is given by l ( x ) = C x - 2ax . Find (a) the total charge
(
)
q in terms of C , and (b) the electric potential at the following points: the origin, the point x = 2a , the
point x = 3a . Express your answers in (b) in terms of q .
3. A circular disk of radius a lies on the x-y plane with its center at the origin. It carries a surface charge
with density given by s ( r ) = C a 2 - r 2 where r is the distance from the center and C is a constant.
(
)
Find (a) the total charge q , and (b) the electric potential at the point ( 0, 0, z ) on the z-axis. Express your
answers in (b) in terms of q .
4. The electric field in a region between the planes x = 0 and x = a has only x-component and is given by
æ x2 ö
Ex = E0 ç1- 2 ÷ where Find the potential difference V (a)  V (0) .
è a ø
5. The electric potential in a certain region as a function of the x-coordinate (in meters) is
V x   4  6 x  7 x 2 (V ) . Find the x-component of the electric field at the point x=2.
6. The electric potential between two metallic plates in a diode is given by V ( x)  Ax 4 3 where x is the
distance in meters measured from one plate. The distance between the plates is 2.0cm, and the potential
at the other plate is 1000V. Determine the constant A and the electric field at the point x  1.0cm
7. On the x-y plane where coordinates are measured in meters, the electric potential at the point (x,y) is
given by
V ( x, y)  8x 2 y  3 y 3 (V )
Find the x any y components of the electric field at the point (2,-1).
8. On the x-y plane, two point charges q are located at the points (a, 0) and (-a, 0) . Write an expression for
the electric potential at the point (x, y) . Use it to find the x and y components of the electric field at the
point (0,y) on the y-axis. Hence determine the y coordinate of the point on the y=axis where the electric
field is maximum.
Answers:
1. 5Ca3 6, 9kq 5a, ( 6 5) ( 6ln2 - 7 / 2) kq a, 3kq 5a
2. -4Ca3 3, 3kq 2a,
3. p a 4C 2,
(8kq
3kq 2a,
[3- (9 / 4)ln3] kq
a
32
é
ù
3a) ê(1+ z 2 a 2 ) - z 3 a3 - 3z 2aú
ë
û
4. -2E0 a 3
5. -22V m
6. 1.84 ´105, - 5.29´10 4 V m
7. 32V m, - 23V m
æ
ö
1
1
ç
÷, 0, 2kqy
8. kq
+
2
2
ç
÷
2
y2 + a2
( x + a) + y 2 ø
è ( x - a) + y
(
)
32
2
,
a
2