**AP Physics – Electric Flux and
Potential Example Problems**

1. There
is a uniform electric field for regions near the surface of the earth equal to -150 N/C **k**. (It is directed toward
the center of the earth or simply “downward”.) Determine the electric flux for
each of the following: (a) the foor of the room, dimensions = 9.0 m × 14 m,
(b) a wall of the room, dimensions = 14 m × 3.0 m, (c) a book cover of
dimensions 9.0 cm × 14 cm that is tilted open at a 45° angle as the book rests on a table, and (d) a sheet stretched
tight with each corner attached to one of the four corners of the room (at any
height) such that the edges of the sheet abut the walls of the room.

2. Supposing the earth’s electric field “obeys” an “inverse square law”, determine the electric flux for the same floor as above located at an altitude 0.414 times the radius of the earth (with its surface level).

3. A shoe box of dimensions 30.0 cm × 15.0 cm × 10.0 cm is tilted with one of its edges touching the floor. Determine the net electric flux for the box and show that it does not depend upon the angle at which it is tilted. Note: the net electric flux is the sum of the flux values for all six sides of the box.

4. An
electric field of variable strength is described by: **E** = *c y*^{3}
**k** within a certain volume of space. Determine an expression for the
electric flux through a rectangle with corners at coordinates: (0, *a*,
0), (0, *b*, 0), (*w*, *a*, 0), and (*w*, *b*, 0).
Under what conditions would the flux be zero? negative? positive?

5. A
cylinder of length *L* and radius *R* is centered on the *z*-axis
in a region where there is a uniform electric field of *E* **i**.
Determine the flux for the fourth of the cylindrical surface where *x* > 0 and *y* > 0.

6. Determine
the electric flux for a cylindrical surface of length *L* and radius *R*
centered along an infinite line of charge with charge per length = λ.
(Note: the electric field is given by **E** = λ/(2πε_{o}*r*)
**r**.)

7. A
line of charge λ is located on the *z*-axis. Determine the electric
flux for a rectangular surface with corners at coordinates: (0, *R*, 0),
(*w*, *R*, 0), (*0*, *R*, *L*), and (*w*, *R*,
*L*).

8. A
point charge *q* is located at the origin. A disk of radius *R* that
is parallel to the *xy* plane is located with its center at (0, 0, *z*).
Determine the electric flux for the disk.

9. A
point charge *q* is located at the origin. A spherical surface of radius *R*
is centered on the origin. Determine the electric flux for the surface.

10. A point charge *q* = −3.0
nC is located at the precise center of a tetrahedron. (a) Determine the
electric flux through one face of the tetrahedron. Now suppose a second charge
of *q* = +2.0 nC is located at the precise center of a second tetrahedron
that shares a face with the first. (b) Find the flux through the shared face.
(c) Find the net flux for the entire outer surface of the combined
tetrahedrons. (d) Can you determine the flux of one of the faces that is __not__
shared? Discuss.

11. A nonconducting sphere of
radius *R* has a charge *Q* distributed uniformly through its
volume. Determine the electric field.

12. Use Gauss’s Law to determine the electric field of a line of charge λ.

13. A cylinder of radius *R*
and infinite length has a uniform charge density *ρ*. Determine the
electric field.

14. Use Gauss’s Law to determine the electric field of a sheet of charge σ.

15. A conducting plate has a uniform charge per area σ distributed along its surface. Determine the electric field inside the plate and immediately outside the plate.

16. A point charge of +3.0 nC is
surrounded by a spherical uniform surface charge of -4.0 nC with radius 20.0 cm. Determine the electric field at *r*
= 10.0 cm and *r* = 30.0 cm. Repeat for a case where the spherical charge
is -3.0 nC instead of -4.0 nC.

17. An infinitely wide and long
nonconducting plate has a uniform charge density *ρ*. Determine the
electric field.

18. A nonconducting sphere has a
*nonuniform* charge per volume given by: *ρ* = *ar* + *b*,
where *a* and *b* are constants. Determine the electric field.

19. What would be Gauss’s Law
for gravitation? Use this result and the previous problem to determine an
equation for *g* that applies to the earth’s interior. Determine the
values of *a* and *b* given that the density at the center of the
earth is estimated to be 12000 kg/m^{3}.

20. Two point charges are
separated by 2__0__ cm. The charges have values: *q*_{1} = -2.0 nC and *q*_{2} = -3.0 nC. (a) Determine the potential energy
relative to infinity. (b) Determine the amount of work necessary to move *q*_{2}
closer so that it is only 15 cm away from *q*_{1}.

21. A point charge *q*_{1}
= -6.0 nC is separated by 5.0 cm from
another point charge, *q*_{2} = +1.0 nC. (a) Determine the amount
of work necessary to separate the two charges to a great distance. (b)
Determine the work necessary to separate the two charges to a distance of 5.0
m.

22. Determine the amount of work
necessary to assemble three point charges, *q*, *q*, and -*q*, into an equilateral triangle of
side *a*.

23. Two deuterium nuclei in the
core of the Sun head toward one another, moving in exactly opposite directions.
In order for fusion to occur the nuclei must be separated by less than 5.0 × 10^{−15}
m. Assuming each nucleus is moving equally fast what must be the speed of each
in order for fusion to occur? Each nucleus has mass 3.34 × 10^{−27}
kg and charge 1.60 × 10^{−19} C.

24. In the simple Bohr model of
the hydrogen atom, an electron moves in a circular orbit of radius *r*
around a fixed proton. (a) What is the potential energy of the electron? (b)
What is the kinetic energy of the electron? (c) Calculate the total energy
when it is in its ground state with an orbital radius of *r* = 5.30 × 10^{-11} m. (d) How much energy is required
to ionize the atom from its ground state?

25. A pith ball of mass 0.10 grams is brought near a charged sphere of radius 15 cm and charge −2.0 μC. The pith ball touches the sphere and acquires a charge of −1.0 nC and quickly darts away. Ignoring gravity estimate the speed of the pith ball at a distance of 5 cm from the surface of the sphere.

26. (a) Determine the electric potential at distances 1.0 cm, 2.0 cm, and 3.0 cm from a point charge of +5.0 nC. (b) How much potential energy would a +10.0 nC charge have if it is placed 3.0 cm from the +5.0 nC charge?

27. A point charge of +*q*
is located at the origin. A second charge of -2*q*
is located at coordinates (*a*, 0). (a) Determine the electric potential
at (*a*/2, 0). (b) Determine the electric potential at (*a*/4, 0).
(c) Determine the electric potential at (0, *a*). (d) Determine the
point(s) where *V* = 0. (e) Determine the point(s) where **E** = 0.
Sketch the electric field and equipotential lines.

28. A charge *Q* is
uniformly distributed in a ring of radius *R* centered on the origin and
located in the *xy* plane. Determine the potential for an arbitrary point
along the *z*-axis.

29. Repeat the previous problem but with a disk of charge instead of a ring of charge.

30. Determine the potential difference from a distance of 1.0 cm to a distance of 3.0 cm from a point charge of - 5.0 nC. (Hint: Use the results from one of the previous problems.)

31. (a) Given the electric field
of the earth: -150 N/C **k**, determine
the potential difference from the floor to the ceiling, a distance of 3.0 m. (b)
As a positive charge moves from the floor to the ceiling does it gain or lose
potential energy? (c) As a positive charge moves from the floor to the ceiling
does the electric field do positive or negative work? (d) How about a negative
charge?

32. The electric field of an
infinite line of charge is given by **E** = λ/(2πε_{o}*r*)
**r**. Use this to estimate the potential difference from 0.5 cm to 1.5 cm
away from the center of a plastic rod of length 20.0 cm and charge -6.0 nC.

33. A proton initially at the
origin with velocity 3.0 Mm/s **i** encounters a region with electric potential
described by *V* = (12000 V) + (5000 V/m)*x*. (a) Determine the
kinetic energy and velocity of the proton as it reaches *x* = 5.0 m. (b)
How far will the proton “penetrate” this region? (c) Determine the electric
field in this region. (d) Repeat part (a) for an electron under the same
conditions.

34. Two parallel aluminum square plates (10.2 cm on a side) are separated by 2.0 mm and connected to a 6.0 V battery. (a) Sketch the electric field and equipotential lines. (b) Determine the electric field between the plates. (c) Determine the charge density on each plate. (d) Determine the amount of charge on each plate. (e) Repeat these steps for the same plates separated by 6.0 mm.

35. Suppose a CRT requires
electrons to be accelerated from rest to 7.0 × 10^{7} m/s. This
acceleration occurs between two parallel plates separated by 0.50 cm. (a)
Determine the required voltage difference across the plates. (b) Determine the
electric field between the plates. (c) Determine the surface charge density on
each plate.

36. Use the result from problem #27 to determine the electric field along the axis of a ring of charge.

37. The electric potential in a
certain region is given by *V* = 3*xy*. Determine algebraic
expressions for the components of the electric field in the same region.
Sketch the electric field and equipotential lines.

38. The Van de Graff generator
has a sphere of diameter 23 cm and a charge of -2.0
μC.

(a) Determine the electric potential at its surface. (b) Determine the amount
of work that must be done to produce that charge on the sphere. (c) Determine
the maximum charge and electric potential that would be possible with a sphere
of this diameter given the dielectric strength of the air is 3.0 × 10^{6}
V/m.

39. Now suppose you consider that the negative charge placed on the sphere of the generator is being transferred not “from infinity” but from a nearby sphere of equal size (and, eventually, equal but opposite charge). Discuss qualitatively how the results of the previous problem would be different.

40. The 30.0 cm diameter metal sphere atop a Van de Graaff generator is connected by a long wire to a second metal sphere of diameter 10.0 cm. If the larger sphere obtains net charge +3.0 μC, what is the charge on the smaller sphere? Find the charge per area for each sphere.

41. An “infinite” metal pipe
with zero net charge has inner radius *a* and outer radius *b*. An
“infinite” line of charge λ is inserted along the axis of the pipe. (a)
Determine the surface charge densities for the inner surface of the pipe and
for the outer surface of the pipe.

(b) Determine the electric field for regions inside, within, and outside the
pipe.

(c) Determine the potential difference from *r* = *a*/2 to *r* =
2*b*. (d) Determine the electric potential (relative to infinity) of the
inside surface of the pipe.

42. A uniformly charged nonconducting sphere of radius 10.0 cm and charge -2.0 μC is surrounded by a thin conducting shell of radius 60.0 cm and charge +3.0 μC. Determine the potential difference between the center of the sphere and the shell. Determine the potential at the center of the sphere (relative to infinity).

Answers

1. a.
−19000 N m^{2}/C

b. 0

c. −1.3 N m^{2}/C

d. −19000 N m^{2}/C

2. −4700
N m^{2}/C

3. 0

4. _{}

5. *ELR*

6. _{}

7. _{}

8. _{}

9. _{}

10. a.

b.

c.

d.

11. _{}

_{}

12. _{}

13. _{}

_{}

14. _{} perpendicular to surface

15. _{} perpendicular to surface

16. a. 2700 N/C away from
center; 100 N/C toward center

b. 2700 N/C away from center; 0

17. _{}, perpendicular to surface for
all points outside the slab

_{},
perpendicular to surface for all points inside the slab

18. _{}

_{}

19. _{}

_{}

*a* = −1.36 × 10^{−3} kg/m^{4} and *b* =
12000 kg/m^{3}

20. a. 2.7 × 10^{−7}
J

b. 9.0 × 10^{−8} J

21. a. 1.1 × 10^{−6}
J

b. 1.1 × 10^{−6} J

22. _{}

23. 3.7 × 10^{6} m/s

24. a. _{}

b. _{}

c. −2.18 × 10^{−18} J

d. 2.18 × 10^{−18} J (13.6 eV as expected)

25. 0.77 m/s

26. a. 4500 V, 2250 V, 1500 V

b. 15 μJ

27. a. _{}

b. _{}

c. _{}

d. *V* = 0 for all points that satisfy: _{}

(This is a circle of radius 2*a*/3 centered on (−*a*/3, 0)

e.

28. _{}

29. _{}

30. 3000 V

31. a. 450 V

b. gain

c. negative

d. lose; positive

32. 590 V

33. a. *v* = 2.0 Mm/s **i**

b. 9.4 m

c. −5000 V/m **i**

d. 93.7 Mm/s

34. a.

b. 3000 V/m

c. 2.66 × 10^{−8} C/m^{2}

d. 0.28 nC

e. 1000 V/m, 8.85 × 10^{−9} C/m^{2}, 0.092 nC

35. a. 13.9 kV

b. 279 kN/C

c. 2.47 × 10^{−6} C/m^{2}

36. _{}

37. **E** = −3*y* **i**
−3*x* **j**

38. a. −156 kV

b. 0.156 J

c. 4.4 μC

39.

40.

41. a. _{} , _{}

b. _{}

_{}

c.

42. a. Shell is at potential 240
kV greater than center

b. −225 kV