Download PHYS 196 Class Problem 1

PHYS 196 Home Work 5
1. 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.
2. 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
3. The electric field inside a diode is given by Ex  Cx1 3 where x is the distance measured from one
plate. The other plate is at a distance a away. Find the potential difference V (a)  V (0) .
0
a
x

4. The electric field on the x-axis due to a point charge fixed at the origin is given by E  b x 2 iˆ where
b  2.0kV  m and x  0 . (a) Find the magnitude and sign of the point charge. (b) Find the potential
difference between the point on the x axis at x  1.00m and x  2.00m . Which of these points is at the
higher potential?
5. A charge of  2.00nC is uniformly distributed on a ring of radius 10.0cm that lies in the x  0 plane and
is centered at the origin. A point charge of  1.00nC is initially located on the x-axis at x  50.0cm .
Find the work required to move the point charge to the origin.
6. A semi-circular arc carries a uniform linear charge distribution of 5.0nC / m . Find the potential at the
center.
7. (Difficult)Two rings of equal radius r are on the planes x  a and x  a with centers on the x-axis.
They both carry total charge Q , uniformly distributed. Find an expression for V x , the potential at the
point x,0,0 and approximate it for x  a. Determine the x-component of the electric field at this
point. Hence find the period of simple harmonic motion executed by a small particle with charge q and
mass m about the origin.
r
a
x
8. 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?
9. 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?
1
10. 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 .
11. 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?
12. A spherical conductor of radius R1 is charged to 20 kV. When it is connected by a long, very thin
conducting wire o a second conducting sphere far away, its potential drops to 12 kV. What is the radius
of the second sphere?
13. 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.
14. A circular disk of radius R has a surface charge distribution given by    0 r 2 R 2 where  0 is a
constant and r is the distance from the center. (a) Find the total charge on the disk. (b) Find an
expression for the electric potential at a distance z from the center of the disk and on the axis of the disk.
z
15. A circle of radius a is removed from the center of a uniformly charged circular disk of radius b and
charge per unit area  , thus forming a flat ring. Find the potential at a point on the axis at a distance
z from the ring’s center. Find also an approximate formula for the potential when z is very large.
16. A long cylindrical shell of length L and radius a is surrounded by a concentric cylindrical shell of radius
b and the same length. The inner shell carries a total charge  q while the larger shell carries  q . Find
the potential difference Va  Vb between the two shells.
17. 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.
18. 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?
19. (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 
20. 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?
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Answers:
1.  22V / m
2.  53kV / m
3. 3Ca 4 3 4
4.  1.0kV
5. 150nJ
6. 140V
52


1
1
kQ 
2a 2  r 2 2 
m a2  r 2


7. V  kQ


2 2
x , T  2
2 
2
2 
2
 r 2  a  x 2
r  a 2 
2kQq 2a 2  r 2
r

a



r

a

x


8. 6.25kV / m, 0kV / m, 750V , 750V , 1.56kV / m, 375V
9. 600kV
10. (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
11. Q 3
12. R2  2R1 3
13. 11kQ 8R
4kQ
R 2  z 2 R 2  2 z 2   2 z 3
14.
4
3R
2kQ
kQ
z 2  b2  z 2  a 2 ,
15. 2
2
b a
z
q
b
ln
16.
20 L a
17. 1.44MeV
18.  9mJ
19.
20. 250W




3



