Physics 2220 Fall 2010

George Williams

FIRST MIDTERM - REVIEW PROBLEMS

A data sheet is provided (last page). You will find some problems involving gravity in the solution set. Ignore them.

The last four problems are from the first midterm exam of 2006. Solutions for problems 13, 36, 41-43 and 111 are not

available. Note the integrals on the data sheet. Any integral you may need (other than polynomials, will be given).

1. (a) Calculate the magnitude of the electric field 12.5 m from a point charge of value +18.7 × 10 C. -3

(b) Calculate the electric force between a proton and an electron that are 0.50 × 10 m apart. (The proton-10

charge is equal in magnitude to the electron charge, but opposite in sign.)

(c) Calculate the first three terms of the binomial expansion for the expression below, for a << x.

1/(x + a ) . 2 2 +7/2

2. Two charges of equal magnitude, +Q and -Q are a distance a apart on the

x-axis. Take x = 0 at the midpoint between the two charges.

(a) Write a general expression for the electric field at point P, at the

coordinate x.

(b) Using the binomial expansion, calculate the electric field at P,

keeping the first two non-zero terms involving a.

(c) If a/x is 0.10, what fractional error is introduced by ignoring the

last term in part (b) (that is, what fraction

of the total answer is this last term)?

1 13. A uniformly charged sphere of a nonconductor has a radius R and charge Q . It is enclosed in a concentric thin

2 2 1 2metal spherical shell whose radius is R . (R > R .) The metal shell has total charge Q .

(a) Calculate the electric field a distance 37.0 cm from the common center of the two

spheres. (Numerical answer including sign.)

(b) Calculate the electric field a distance of 17.5 cm from the common center.

(Numerical answer including signs.)

(c) Calculate the electric field a distance 2.00 cm from the common center.

(Numerical answer including sign.)

1 2Q = +175 :C; Q = -325 :C

1 2R = 10.0 cm; R = 25.0 cm

4. (a) Calculate the electric force between two positive charges of 6.0 :C and 9.0 :C a distance of 3 cm apart.

5. (b) Find the force between two electrons 1.0 × 10 m apart.9

(d) Find the electric field 2.0 × 10 m away from a helium nucleus with charge +2e.-10

(Note: Error in exam solution. Should have been 10 .)-10

6. (e) Find the electric field 20.0 m away from a metal sphere which has a charge of 3.0 × 10 C. 9

7. (a) Calculate the electric force between a lithium nucleus (z = 3) and an electron a distance 1.00 × 10 m-11

away.

(c) Calculate the electric field magnitude at a point midway between a

charge of +7.0 pC and one of -3.5 pC. The distance is 0.75 m.

1 28. Two charges q and q when combined give a total charge of 6.00 :C. When they are separated by 3.00 m, the

force exerted by one charge on the other has the magnitude of 8.00 × 10 N. -3

1 2(a) Find q and q if both are positive so that they repel each other.

1 2(b) Find q and q if one is positive and the other negative so that they attract each other.

9. Given the square charge distribution pictured. Find the force on the charge at A

(magnitude and direction).

10. Three charges of equal magnitude are arranged at the corners of an equilateral

triangle of side a as shown.

(a) Find the electric field at point A, the midpoint of the top side. Use

the coordinate system shown.

(b) Find the force, magnitude and direction, on a charge -q placed at A.

11. Given the charge arrangement shown.

(a) Find the electric field at point P. Express this using the unit vectors

i and j.

(b) Find the magnitude of the force on a charge of +3q placed at P.

12. An electron is placed at each corner of an equilateral triangle having sides of length 20 cm.

(a) What is the electric field at the midpoint of one of the sides?

(b) What force would another electron placed there experience?

13. Given a quarter circle of radius R that is uniformly charged with a total charge -Q,

find the electric field (magnitude and direction) at point P, the center of the circle.

14. A thin wire carries a uniform charge density 8, and

ois bent into a circular arc that subtends an angle 22 as shown in the figure. Find

the electric field at the center of the arc.

15. Given a uniformly charged, nonconducting rod of length 2a. The total charge on

the rod is Q. Find the electric field (magnitude and direction) at point P, a

distance a from the left end of the rod. Use the coordinate system shown.

16. A spherically symmetric charge distribution of total charge +Q can be described by the relation

1 2 2D(charge density) = A/r for R < r < R , D = 0 everywhere else.3

(a) Find the value of the constant A.

1 2(b) Find the value of the electric field at any value of r between R and R .

17. A cylindrically symmetric charge distribution in a nonconductor is described by the relation D = A/r for 3

1 2R < r < R and is zero everywhere else. (A is a constant.)

(a) Find the total charge on a length L of the cylinder.

1 2(b) Find the electric field at a point P, where P is between R and R .

18. (a) Use the binomial expansion to calculate the coefficient of x for (1 + x) .3 -9/2

(c) Calculate the magnitude of the electric field at a distance of 1.00 × 10 m from a proton. (The charge on-10

the proton is +e = +1.6 × 10 C.) -19

(e) A fine wire 12.0 m long has a total charge of +4.25 × 10 C. Calculate the electric field, magnitude and-8

direction 1.00 mm away from the center of the wire at a point halfway between the ends.

19. In the diagram shown |q| = 3.00 × 10 C. The point P is directly below the-12

negative charge. Take a = 2.00 × 10 m.-4

(a) Calculate the electric field, magnitude and direction at point P.

(b) Calculate the magnitude of the force on a charge Q = 4.00 × 10 C-7

placed at P.

1 220. A thick hollow sphere is constructed with inner radius R and outer radius R .

1 2Between R and R there is a charge density given by D = A/R, where A is constant.

Everywhere else there is no charge.

P 2(a) Calculate the electric field at point P, where R > R .

1 Q 2(b) Calculate the electric field at point Q, where R < R < R .

21. Three charges are at the corners of an equilateral triangle as shown.

(a) Calculate the electric field (magnitude and direction) at point P.

(b) Calculate the force on a charge -4Q (magnitude and direction) placed

at P.

o o22. Consider a sphere of nonconductor which has a charge density given by D = D (1 - "R ), where D and " are3

oconstants. The sphere has a radius of R .

(a) Calculate the magnitude of the electric field an arbitrary distance R from the center of the sphere where R

o o o< R . (Leave this answer in symbolic form in terms of D , ", R , R, k as needed.)

(b) If the sphere has a radius of 1.25 cm, a total charge of 1.75 × 10 C, and it is known that the charge density-6

goes to zero at the surface, calculate the numerical value of ".

o(c) With the assumptions the same as in (b), calculate the numerical value for D .

23. (a) Calculate the magnitude of the electric force between two electrons that are 0.72 × 10 m apart. -10

(b) What is the magnitude of the acceleration (in m/s ) of an electron in an electric field of magnitude2

275 N/C?

(c) Calculate the magnitude of the electric field at a point midway between

two charges, one of +5.5 × 10 C and the other of -7.5 × 10 C. The-6 -6

distance d is 1.75 m.

(d) If the electric field at the surface of the Earth is found to be 100 N/C, pointed downward, and at 1000 m

above the surface of the Earth is 25 N/C also pointed downward, calculate the number of elementary

charges (e = 1.6 × 10 C) in a cube 1000 m on a side, with its bottom at the Earth's surface. -6

(e) In the binomial expansion of (x + a ) , calculate completely the term in a .2 2 -5/2 4

o o24. A long cylinder of nonconductor of radius R = 1.75 cm, has a charge density given by D(r) = Br for 0 < r < R . 3

The charge density is zero everywhere else.

(a) If the total charge per unit length is -3.75 × 10 C/m, calculate the numerical value of B. -7

(b) Calculate the magnitude of the electric field at a point r = 0.95 cm. State the direction of the field.

25. We have calculated that the field of an electric dipole for points on the axis

oof the dipole is given by |E| = (1/4Bg )(2qa/x ), where 2a is the distance3

between the two charges q, and here the dipole is aligned along the x axis.

An electric quadrupole can be treated as two dipoles, head to head, or tail to

tail, as shown. Using the dimensions given and the binomial expansion,

calculate the first nonzero term for the electric field of the quadrupole shown

for a point +x on the x axis where x >> a.

26. Given a line of charge with charge density 8 = +3.75 :C/m (3.75 × 10 C/m),6

and length R = 1.27 m. Calculate the electric field (magnitude and direction)

at point P located a distance a from the line, and a distance R/4 from one end

of the line. Take a = 0.180 m, and calculate a numerical value complete with

units.

27. (c) Calculate the magnitude of the electric force between two electrons that are 0.2 ×10 m apart. -10

(d) Using the binomial expansion (1 + x ) where x << 1, calculate the coefficient of the term in x .3 -4/3 9

(e) If a cube has a total charge inside of +3.20 nC, calculate the total electric flux, with units, crossing the

faces of the cube.

28. Three charges are place at the vertices of an equilateral triangle as shown.

o(a) Calculate the force, magnitude and direction on a charge q placed at P

which is at the midpoint of the base.

(b) Calculate the electric field at P due to the three charges at the corners of the

otriangle. Q = 9.0 :C; a = 1.25 cm; q = -6.25 :C

29. Two rods of nonconductor are placed at right angles. The rods each have a total

charge of 3.25 × 10 C that is uniformly distributed and their length is L = 1.25 m. -6

Calculate the electric field, magnitude and direction, at point P. P, together with

the ends of rods, form a square of side L.

30. (b) Calculate the magnitude of the electric force between an electron and a proton (the hydrogen nucleus) if

they are 0.75 × 10 m apart. The charge of the proton is positive and is equal in magnitude to the charge-10

of the electron. (Numerical answer.)

(d) Calculate the electric field 2.00 m from the surface of a uniformly charged non-conducting sphere, whose

radius is 3.00 m and whose total charge is 5.75 × 10 C. (Numerical answer.) -3

(e) Use the binomial expansion to calculate the complete term in a for 4

31. Calculate the electric field, both magnitude and direction, at point P.

The direction should be given as an angle measured from the positive x-

axis. (Be sure to include the proper sign for the angle.)

1Q = +125 :C

2Q = -300 :C

3Q = +175 :C

a = 4.25 mm

32. (a) Calculate the magnitude of the electric field at a distance of 0.500 × 10 m (about the size of an atom)!10

from the center of a nucleus with a positive charge of +3e.

(c) For the expression

use the binomial expansion to calculate the term in a . Don't lose any pieces. 6

(e) An electron is accelerated from rest by an electric field of 1.00 × 10 N/C. What is its speed after traveling4

10.0 cm?

33. Three electric charges are placed as shown. Take a = 3.20 × 10 m.-3

(a) Calculate the force, in , notation, on a charge of +9.00 :C

placed at P.

(b) Determine the electric field at P, in , notation.

34. a) For the expression,

calculate the x term in the binomial expansion. 3

(b) An electron is accelerated from rest in an electric field of 2.35 × 10 N/C. Calculate its speed after it has2

traveled 7.00 cm.

(d) Calculate the magnitude of the electric force between an electron and the nucleus of a hydrogen atom. The

nuclear charge is equal to the charge of the electron. Take the distance between them as 1.00 × 10 m. -10

35. Three charges are placed at corners of a rectangle as shown. Point P is at the

4th corner.

(a) Calculate the electric field in notation at point P due to these

charges.

(b) Find the force, magnitude and direction on a electron placed at P. The

direction should be expressed as an angle measured counterclockwise

from the positive x axis.

1 2 3Q = ! 3.00 × 10 C; Q = + 25.00 × 10 C; Q = ! 15.00 × 10 C; a = 8.00 cm-6 -6 -6

36. Points A, B, C and D are located at the corners of a rectangle (see figure).

Charges are placed at A, B and C.

(a) Find the electric field, magnitude and direction, at D (numerical value).

Measure the direction as an angle counterclockwise from the positive x-

axis.

(b) Calculate the force, magnitude and direction a charge QD = -9.00 × 10-6

C, if the charge is at D.

A BQ = +3.20 × 10 C Q = -7.75 × 10 C-6 -6

CQ = +4.45 × 10 C a = 3.45 cm-6

37. (a) Calculate the magnitude of the electric field, including units, 4.75 m from a point charge of value

Q = +2.95 × 10 C. -3

(b) Calculate the complete term that involves a in the binomial expansion of the expression below.4

(e) The nucleus of a lithium atom has an electric charge equal to +3|e|, since it has three protons. Calculate the

magnitude of the electric force between a lithium nucleus and an electron if they are 0.25 × 10 m apart. -10

38. Points A, B and C are at the corners of an equilateral triangle of side a. A

charge of +6.50 × 10 C is placed at B. A charge of -9.00 × 10 C is placed at C. -6 -6

a = 1.25 cm

(a) Calculate the electric field, magnitude and direction, at point A. Angles are

measured counter-clockwise from the positive x-axis.

(b) Calculate the magnitude of the force on a charge of 1.70 × 10 C placed at-6

A.

1 139. A uniformly charged sphere of a non-conductor has radius R and charge Q . It is

2 2enclosed in a concentric thin metal spherical shell whose radius is R (R >

1 2R ). The metal shell has a total charge Q .

(a) Calculate the electric field a distance 37.0 cm from the common center

of the two spheres. Give the answer in numerical form with correct

sign.

(b) Determine the electric field a distance of 17.5 cm from the common

center. Give the answer in numerical form with correct sign.

1 2 -6 1 2Q = +300 × 10 C; Q = -475 × 10 C; R = 8.00 cm; R = 20.0 cm-6

40. (b) Calculate the electric force between one electron and a lithium nucleus if they are 1.70 × 10 m apart. -11

The lithium nucleus has a charge of +3e.

(d) Calculate the complete term involving x for the function (1 - x) , using the binomial expansion. 4 -4/3

41. Given the arrangement of charges shown.

(a) Calculate the magnitude of the electrostatic force on number 4 due to 1, 2

and 3.

(b) Determine the direction of the electrostatic force on 4 due to 1, 2 and 3.

Express as an angle positive counterclockwise.

(c) Find the magnitude of the electric field at 4 due to 1, 2 and 3. Don’t forget

units.

1 2Q = -6.20 × 10 C Q = +8.30 × 10 C-6 -6

3 4Q = -11.0 × 10 C Q = +4.20 × 10 C-6 -6

a = 3.20 × 10 m-4

42. A thin rod of length L is uniformly charged with a negative

charge. The total charge is !Q. Determine the x-component of the

electric field at point P, a distance a from the end of the rod. a is

perpendicular to the rod. Be sure to include the sign of your

answer.

43. (a) A lithium-7 nucleus has 3 protons and 4 neutrons. Calculate the force between a lithium nucleus and an

electron 3.00 × 10 m apart. -11

(b) Calculate the electric field (in N/C) at a distance of 4.00 × 10 m from a lithium nucleus. -9

(c) A conducting sphere of radius 1.50 × 10 m has a positive charge of 1.75 × 10 C. Determine the electric-2 -11

field at a distance of 3.30 m from the center of the sphere.

(d) For the expression 1/(x - a) , where a << x, calculate completely the term in a using the binomial35/2

expansion.

(e) An electron is accelerated from rest in a uniform electric field of 4.67 N/C. Find the speed of the electron

after it has traveled 3.25 cm.

44. Three charges are placed at the corners of a square as shown.

[Q = 1.70 × 10 C]-10

(a) Determine the electric field at point P in notation (numerical

value).

(b) Calculate the magnitude of the electric field at point P (numerical

value).

a = 1.50 x 10 m--6

45. The curved object is a thin rod bent in a circular form that goes 3/4 of the way around

the circle from the positive x-axis to the negative y-axis. The origin is at the

center of the circle. There is a uniform linear charge density and a total charge

Q = 4.70 × 10 C. Find the electric field, magnitude and direction, at the origin. -9

Ignore the diameter of the rod itself.

o46. A sphere of non-conductor has a charge density given by D(r) = Br , for r < R ,3

oand zero everywhere outside of R .

(a) Find the total charge on the sphere.

(b) Calculate the electric field at point P, a distance R from the center

1 o(R > R ).

2(c) Determine the electric field at point Q, a distance R from the center, where

2 oR < R .

47. (a) Calculate the force between two protons (q = +1.6 × 10 C) that are 1.5 × 10 m apart (approximately!19 !15

the separation that exists in an atomic nucleus).

(b) For the expression (x ! a) , where a << x, calculate completely the term in a using the binomial!7/2 2

expansion.

(c) An electron is accelerated from rest in an electric field of 0.325 N/C. After it has traveled 1.20 m, what is

its speed?

(d) Calculate the numerical magnitude of the electric field at a distance of 1.20 × 10 m from an electron. !10

(e) A cube of non-conductor is uniformly charged with a total charge of +6.20 × 10 C. Calculate the electric!6

flux through one face of the cube.

48. The points shown are at the corners of a square.

(a) Calculate the electric field, magnitude and direction, at point P.

(b) Calculate the force, magnitude and direction, on a negative charge of

!5 Q placed at P.

49. A spherically symmetric charge distribution on a non-conductor is modeled by a charge density given by:

,

o owhere B is a constant. The charge distribution has an outer radius R . There is no charge outside of R .

(a) Calculate the total charge Q.

o(b) Calculate the electric field at the interior point r = R /3.

50. A thin rod of length L is uniformly charged with a negative charge.

The total charge on the rod is !Q. Determine the y-component of the

electric field at point P, a distance a from the end of the rod. a is

perpendicular to the rod. Be sure to include the sign of your answer.

51. (a) Calculate the magnitude of the electric field 4.00 × 10 m away from a proton, which has a positive- 12

charge identical in magnitude to the electron.

(b) Calculate the term in x for the binomial expansion of (1 - x ) . 6 2 - 9/2

(c) Calculate the electric force between two protons a distance 2.00 × 10 meters apart. - 13

(d) An electron is accelerated from rest in a uniform electric field of 175 N/C. Calculate its speed after it

travels 1.50 m parallel to this field.

(e) A cube of non-conductor is uniformly charged with 9.30 × 10 C. Calculate the electric flux through one- 5

face of the cube.

152. Q = +2.40 :C

2Q = -1.75 :C

3Q = +3.00 :C

(a) Calculate the electric force, magnitude and direction, on a charge of +2.00 :C

placed at point P. Calculate the direction as an angle measured from the positive

x-axis with counterclockwise positive. [The usual convention.]

(b) Calculate the electric field, with proper units, magnitude and direction, at point P.

153. A charge of + 6.00 :C is placed on a spherical, conducting shell whose inner radius is R

2and outer radius is R . Then another charge of -9.00 :C is placed at the exact center of the

spherical shell.

2(a) Calculate the electric field at a distance R from the center if R > R .

1 2(b) Calculate the electric field at a distance R from the center if R < R < R .

1(c) Calculate the electric field a distance R from the center if R < R .

o54. A long non-conducting cylinder of radius R has a volume charge density given by D = Br , where B is a constant. 4

oD = 0 for r > R (outside the cylinder).

(a) Calculate the total charge in a length L of this cylinder.

1 1 o(b) Calculate the electric field a distance R away from the axis of the cylinder if R > R .

2 2 o(c) Calculate the electric field a distance R away from the axis of the cylinder if R < R .

55. (a) Calculate the magnitude of the electric field at a distance of 2.00 × 10 m from a proton -10

proton(Q = +1.6 × 10 C). -19

(b) Calculate the magnitude of the electric force between two electrons that are 0.60 × 10 m apart. -10

(c) Calculate the magnitude of the electric field at point P, half-way

between the two charges.

(d) Calculate the coefficient of x for the binomial expansion of (x << 1): (1 - x ) . 3 2 -7/2

(e) A cube of non-conducting material is uniformly charged with a total charge of 7.50 × 10 C. Calculate the-6

electric flux through one face of the cube.

56. Calculate the electric field, magnitude and direction, at the center of the

square of charges. Express direction as an angle from the positive x-axis.

Positive angles are counterclockwise from x-axis.

1Q = +5.70 nC

2Q = +2.75 nC

3Q = -7.50 nC

4Q = +13.2 nC

a = 4.20 × 10 m-4

57. A thick, hollow sphere is constructed out of non-conducting material. The inner

1 2. 1 2radius is R and the outer radius is R . The charge density between R and R can

be expressed as D = A/R , where A is a constant. [R is the variable.]2

P 2(a) Determine the electric field at point P, where R > R .

2 Q 1(b) Find the electric field at point Q, where R > R > R .

58. Given a uniformly positively charged rod of total length 2L

and total charge Q. Calculate the electric field at point P,

magnitude and direction, due to the rod.

59. Two sides of a square of nonconductor of side length a are positively charged with a

linear charge density of 3.70 × 10 C/m.-9

(a) Calculate the electric potential at point P. P is at the corner of the rectangle.

Both the rectangle and P are in the plane of the paper. (A numerical answer, in

volts, is required.)

(b) Calculate the work to bring a proton from infinity to P. (The proton is the

nucleus of the hydrogen atom. It has a positive charge equal in magnitude to the

charge on the electron.)

60. (a) What is the surface charge density of a conducting sphere of radius R, if the total charge on the sphere is

Q?

(b) Calculate the work to move 6.00 :C of charge through a potential difference of 60.0 volts.

o(c) If the potential is given by V = V (1 - "x ), find the x component of the electric field. 2

(d) In moving a charge of 0.20 C from point A to point B, 350 Joules of work is done. Calculate the potential

difference between A and B.

(e) Express 3.20 × 10 Joules in electron volts. -18

61. Two metal spheres are 3.0 cm in radius and carry charges of +1.0 × 10 C and -3.0 × 10 C, respectively,-8 -8

assumed to be uniformly distributed. There centers are 2.0 m apart.

(a) Calculate the potential of the point halfway between their centers.

(b) Find the potential of each sphere.

62. On a thin rod of length L lying along the x-axis with one end at the origin (x = 0),

as shown in the figure, there is distributed a charge per unit length given by 8 =

kx, where k is a constant.

(a) Taking the electrostatic potential at infinity to be zero, find V at point P on

the y-axis. P is a distance a from the origin.

y(b) Determine the vertical component, E , of the electric field intensity at P

from the result of part (a).

63. (a) Calculate the electric potential at point P in the drawing shown.

(b) Calculate the magnitude and direction of the electric field at point P. Use

the coordinate system shown.

1Q = +6.50 :C

2Q = -7.50 :C

3Q = +9.75 :C

64. Three charges are placed at the corners of an equilateral triangle of side a as

shown. Use the coordinate system shown. Take the origin at P.

(a) Find the electric field (magnitude and direction) at point P, the midpoint of

the base.

(b) Find the electric potential at point P.

65. Four equal charges are arranged in a square of side a as shown.

(a) Find the electric potential at point A which is at the midpoint of the side.

(b) Find the work to bring a charge of -Q from infinity to point A.

66. Given a nonconducting rod of charge whose charge density is +8 C/m which begins at x = 0 and extends to

infinity along the positive x axis. Calculate the electric field at point x = -a. (Include its direction.)

67. A cylindrical nonconductor has a charge distribution given by a volume charge density

o o oexpressed by D = D (1 - "R ) for R < R , D = 0 for R > R (" is a constant).2

(a) Find the magnitude of the electric field at P, a distance R from the center of cylinder.

State its direction in words.

o(b) Find the electric field at a radius R = R /4.

68. Given a long cylinder of a nonconducting material. This cylinder is charged with

o8 = 2.75 × 10 C/m. The volume charge density is given by D = D (1 - "R ) where the charge density goes to-8 3/2

o ozero at R = R = 3.00 cm, the outside surface of the rod. If " and D are constants, find:

o(a) numerical values for " and D ,

o(b) the electric field at R = 3/4 R , and

(c) the electric potential difference between the center and outside surface of the rod.

69. Given an electric dipole consisting of a charge q = +3.75 :C a distance 2a from a charge of q = -

3.75 :C. Take a = 1.00 mm. Use the binomial theorem to calculate the electric potential at a

distance R of 3.75 m from the center of the dipole along the axis of the dipole.

o70. Consider a cylinder of nonconductor whose radius is R and whose length is infinite with a

volume charge density D = AR where R is the distance from the center of the cylinder. 2

Calculate the electric potential difference between a point P and the surface of the cylinder.

oP is at a value of R = 2/3 R . Assume the cylinder is positively charged and state clearly the

osign of the potential difference V(P) - V(R ).

71. Use the binomial theorem to calculate the electric potential at point P a

distance R from the center of an electric quadrupole. The quadrupole has

charges +Q at distances ±a from a charge -2Q. The point P is on the axis of

the quadrupole. Assume a << R. Calculate the first two nonzero terms

involving a.

72. A sphere of nonconductor is negatively charged with a charge density given by D = AR ,3

owhere A is a constant. The radius of the sphere is R .

(a) Calculate the magnitude of the electric potential difference between the surface of

othe sphere and a point P, a distance P from the center. (R < R .)

o(b) State the sign of the potential difference V(R) - V(R ) and give a clear physical (not

mathematical) reason for this sign.

(c) If the total charge on the sphere is Q, calculate A.

73. Consider a slab of nonconductor with thickness d in the y direction

extending to infinity in the x and z directions. The slab has a uniform

charge density D, where D = 1.45 × 10 C/m . If d = 1.25 m and the-6 3

midpoint is at y = 0, find the electric potential difference between y =

0 and y = +0.45 m.

74. Given a very long, thin wire of radius r and a charge density of +8 c/m.

(a) Calculate the electric potential difference between the surface of the wire, and point at a distance R away

from the center of the wire where R > r.

(b) State clearly the sign of the potential difference, V(R) - V(r), and give a physical reason for it.

o o o75. A sphere of nonconductor has a charge distribution given by D = D (1 - "R ) for R > R , where R is the radius of3

o othe sphere. D = 0 for R > R . The total charge on the sphere is Q. The value of D goes to zero at R = R .

(a) Calculate the electric field at any point within the sphere at a distance R from its center.

o(b) Assume that Q and R are known. Find a formula for ".

o o(c) Assume that Q and R are known. Find a formula for D .

o(d) Calculate the magnitude of electric potential difference between R and R (inside the sphere) assuming the

charge is positive.

76. Four charges are arranged in a line as shown. Calculate the electric

potential at P, in volts, for the values of charges and a given.

1Q = + 2.00 × 10 C-6

2Q = - 3.25 × 10 C-6

3Q = + 4.25 × 10 C-6

4Q = - 8.20 × 10 C-6

a = 3.00 cm

77. A rod of nonconductor has a charge denisty given by 8 = + A#x#. The charge is positive on

either side of x = 0. A is a constant. Calculate the electric potential at point P, a distance a

away from the midpoint of the rod. Take the midoint as x = 0.

1x = -L/2

2x = + L/2

A B78. Given a long hollow tube of inner radius R and outer radius of R . The tube is made

A Bof a nonconductor and has a charge density of D = AR between R and R . D is zero-3

everywhere else.

(a) Calculate the electric field at an arbitrary distance R from the axis of the tube

A Bwhere R < R < R .

A(b) If the potential at R is set at V = 0, calculate the potential difference between a

A A Bpoint at R and a point at an arbitrary value of R where R < R < R .

79. (a) If an electric potential has the form V = - Ax y , calculate the y component of the electric field, including3 4

sign, at the point x = + 5.00, y = + 4.00, z = + 1.00.

(b) There are two protons in the nucleus of a helium atom. If they are initially 1.00 × 10 m apart and-15

released, how much kinetic energy would each have after they have moved 1.00 m apart?

(c) If a conducting sphere of radius 4.00 cm is charged to a potential of 350 V, calculate the surface charge

density on the sphere.

(d) Three charges are arranged as shown. Calculate the electric potential at point P.

1 2 3Q = + 3.00 × 10 C; Q = - 4.70 × 10 C; Q = + 8.50 × 10 C-6 -6 -6

(e) If a conducting rod 6.50 m in length has a charge of +7.50 × 10 C uniformly-8

distributed over its surface, calculate the electric field at the midpoint of the rod, and 2.20 cm from the axis

of the rod. The rod has a radius of 0.500 cm.

o o80. A charged, nonconducting sphere has a charge density given by D = D (1 - AR ), where D and A are constants, the5

total charge on the sphere is 3.30 × 10 C, the radius is 4.00 cm and the charge density goes to zero at its surface. -6

o(a) Calculate the numerical values, with proper units, for D and A.

(b) Find the electric potential at the surface of the sphere (numerical value).

(c) What is the electric field at a point 3.00 cm from the center of the sphere (numerical value)?

81. Four points are at the corners of a square, as shown.

(a) For the values of the charges given, calculate the electric potential

at P.

(b) For the values of the charges given, find the work needed to bring

an electron from far away to P.

(c) For the values of the charges given, find the direction of the electric

field at P, measured as an angle counterclockwise from the positive

x axis.

1 2Q = + 2.70 × 10 C; Q = - 4.00 × 10 C; -7 -6

3Q = +1.00 × 10 C; a = 4.25 mm-7

o o82. A very long cylinder of non-conductor of radius R has a charge density given by D = AR for R < R , and D = 02

ofor R > R . A is a constant.

(a) Using Gauss' Law, calculate the magnitude of the electric field at an arbitrary point within the cylinder a

distance R from the cylinder axis.

o(b) Calculate the magnitude of the potential difference between the wall of the cylinder and its axis [V(R ) -

V(0)].

o(c) If the sign of the charge on the cylinder is negative, state clearly the sign of V(R ) - V(0), and give a

physical reason for it.

83. (a) A wire 27.0 m long carries a total charge of +3.00 × 10 C. If the charge is uniformly distributed (not -12

really true), calculate the electric field 2.00 cm from the center of the wire and at its midpoint. The radius

of the wire is less than 2.00 cm.

(b) A proton (Q = +1.60 × 10 C) is accelerated from rest through a potential difference of 75.0 volts. -19

Calculate its velocity.

(c) An electric potential is described by V = Axy z . Calculate the y component of the electric field at3 2

x = +2.00 m, y = 3.00 m and z = 1.50 m. A is a constant.

(d) Calculate the electric potential at the center of the square (A) for the

array of charges shown.

(e) The electric field just above the surface of conducting sphere is E =

+975 N/C. Calculate the charge density on the surface of the sphere.

84. In the charge distribution shown, P is directly in line with the charges and

is a distance R as shown.

(a) Find an exact expression for the electric potential at P.

(b) Use the binomial expansion to determine the approximate value for

the potential at P keeping only the the first non-zero term in the

answer involving a. (Assume R >> a.)

85. Two conducting spheres are connected with a long wire. Sphere 1 has a

radius of R = 12.0 cm and sphere 2 has a radius of r = 4.00 cm. A charge of

30.0 pC is placed on the system.

(a) What is the charge in picocoulombs on sphere 1?

(b) Find the potential of sphere 1 in volts.

86. A spherical charge density is modeled as D = Ar , where A is a constant and r the distance from the center of the2

osphere. The radius of the sphere is R . (The sign of the charge is included in A.)

o(a) If we take V = 0 at r = 0 (the center of the sphere), find the potential at the point r = R /3.

T o T o(b) If the total charge of the sphere is Q , find the value of A in terms of R , Q and g , as needed.

87. (a) For the arrangement shown, what is the potential difference

V(B) - V(A)?

(b) Use the binomial expansion to calculate the coefficient of x for the expression : 3

with (x << 1).

(c) A very long thin wire has a total charge of +7.0 × 10 C on a length-7

of 30.0 m. Calculate the electric field 0.50 cm away from the center

of the wire, at a point nowhere near its ends.

(d) Calculate a numerical value for the electric field a distance 3.00 × 10 m from an electron. -10

(e) The nucleus of a carbon atom has exactly 6 elementary positive charges. Calculate the potential, in volts, a

distance 1.00 × 10 m from the center of the nucleus. [This is just outside the nucleus.] -14

88. Given the charge distribution shown.

(a) Calculate the electric field, magnitude and direction at point P. Direction must be

clearly shown for full points.

(b) Find the electric potential at point P.

(c) Calculate the worked needed to bring a charge of +4q from infinity to point P.

(d) Calculate the force, magnitude and direction on the +4q charge at P. Direction must be

clearly shown for full points.

o o89. A disk of radius R is charged with a surface charge density given by F = F r. The units of

F are C/m .2

(a) Calculate the total charge on the disk.

(b) Calculate the electric field on the axis of the disk a distance x away from the disk

(out of the paper a distance x).

1 2 1 290. A spherically symmetric shell of non-conductor has an inner radius R and an outer radius R . Between R and R

the charge density is given by . There is NO charge anywhere else.

(a) If the total charge is Q, calculate A.

2 1 2(b) Calculate the electric field at any value of R larger than R in terms of A, R , R , ,

k, R.

1 2 1(c) Determine the electric field at any value of R between R and R in terms of A, R ,

2R ,, R, , k.

2 1(d) What is the sign of V(R ) - V(R ). Explain clearly.

91. (a) Calculate the magnitude of the electric force between two electrons a distance 7.52 × 10 m apart. -11

(b) A conducting sphere of radius 0.175 m is charged with a negative charge of 6.73 × 10 C. Calculate the-6

potential at its surface.

(c) For the arrangement shown, what is the potential difference V(B) - V(A)?

(d) A very long, thin wire has a total charge of Q = +1.57 × 10 C uniformly distributed on its total length of-6

47.2 m. Calculate the magnitude of the electric field a distance 1.75 mm away from the center of the wire

at a point nowhere near its ends. The wire has a radius of 1.00 mm.

(e) Calculate the coefficient involving the x term using the binomial expansion. 12

92. The four points shown are at the corners of a square of side a. Given the values shown,

calculate (numerical values),

(a) the x and y components of the electric field at point P;

(b) the direction of the electric field at point P, measured as an angle counter-

clockwise from the positive x-axis.

1 2 3Q = + 4.76 × 10 C; Q = - 6.35 × 10 C; Q = 2.37 × 10 C; a = 1.35 cm-6 -6 -6

1 193. A uniformly charged sphere of a nonconductor has a radius R and charge Q . It is enclosed

2 2 1in a concentric thin metal spherical shell whose radius is R . (R > R .) The metal shell has

2a net total charge Q .

(a) Calculate the electric field a distance 47.0 cm from the common center of the two

spheres. (Numerical answer including sign.)

(b) Calculate the electric field a distance of 13.5 cm from the common center.

(Numerical answer including signs.)

(c) Calculate the electric field a distance 3.00 cm from the common center. (Numerical

answer including sign.)

1 2 1 2Q = -375 :C; Q = +172 :C; R = 12.0 cm; R = 30.0 cm

o94. A very long cylinder of non-conducting material has a radius R and a volume charge density given by D = BR3

o ofor R < R and D = 0 for R > R . B is a constant.

o(a) Using Gauss’ Law, calculate the electric field a distance R from the axis of the cylinder where R > R .

(b) Using Gauss' Law, calculate the magnitude of the electric field at an arbitrary point within the cylinder a

odistance R from the cylinder axis. R < R

(c) Calculate the magnitude of the potential difference between the wall of the cylinder and its axis

o[V(R ) - V(0)].

o(d) If the sign of the charge on the cylinder is negative, state clearly the sign of V(R ) - V(0), and give a

physical reason for it.

95. (a) Calculate the magnitude of the electric force between two electrons a distance 7.52 × 10 m apart. -11

(b) A conducting sphere of radius 0.175 m is charged with a negative charge of 6.73 × 10 C. Calculate the-6

potential at its surface.

(c) For the arrangement shown, what is the potential difference

V(B) - V(A)?

(d) A very long, thin wire has a total charge of Q = +1.57 × 10 C uniformly distributed on its total length of-6

47.2 m. Calculate the magnitude of the electric field a distance 1.75 mm away from the center of the wire

at a point nowhere near its ends. The wire has a radius of 1.00 mm.

(e) Calculate the coefficient involving the x term using the binomial expansion for 12

96. The four points shown are at the corners of a square of side a. Given the

values shown, calculate (numerical values),

(a) the x and y components of the electric field at point P;

(b) the direction of the electric field at point P, measured as an angle

counter- clockwise from the positive x-axis.

1 2Q = + 4.76 × 10 C Q = - 6.35 × 10 C-6 -6

3Q = 2.37 × 10 C a = 1.35 cm-6

1 197. A uniformly charged sphere of a noncondctor has a radius R and charge Q . It is enclosed in a concentric thin

2 2 1 2metal spherical shell whose radius is R . (R > R .) The metal shell has a net total charge Q .

(a) Calculate the electric field a distance 47.0 cm from the common center of the two

spheres. (Numerical answer including sign.)

(b) Calculate the electric field a distance of 13.5 cm from the common center.

(Numerical answer including signs.)

(c) Calculate the electric field a distance 3.00 cm from the common center.

(Numerical answer including sign.)

1 2 1 2Q = -375 :C; Q = +172 :C; R = 12.0 cm; R = 30.0 cm

o98. A very long cylinder of non-conducting material has a radius R and a volume charge density given by D = BR3

o ofor R < R and D = 0 for R > R . B is a constant.

o(a) Using Gauss’ Law, calculate the electric field a distance R from the axis of the cylinder where R > R .

(b) Using Gauss' Law, calculate the magnitude of the electric field at an arbitrary point within the cylinder a

odistance R from the cylinder axis. R < R

o(c) Calculate the magnitude of the potential difference between the wall of the cylinder and its axis [V(R ) -

V(0)].

o(d) If the sign of the charge on the cylinder is negative, state clearly the sign of V(R ) - V(0), and give a

physical reason for it.

99. (a) Calculate the magnitude of the electric force between two electrons that are 2.00 × 10 m apart. -10

(b) Calculate the electric field a distance 4.32 × 10 m away from the nucleus of a helium atom. The helium-12

atom nucleus has 2 protons and 2 neutrons.

(c) Find the point on the x-axis, between the two charges, where the

electric potential is zero.

(d) A proton is accelerated from rest through an electric potential difference of 5320 volts. Find its final

velocity.

(e) A very long wire has a linear charge density 8, of 275 × 10 C/m. Calculate the magnitude of the electric-12

field a distance of 3.75 cm from the center of the wire. (This is outside of the wire.)

1 2 3100. Charges Q , Q , and Q are at three corners of a square of side a.

(a) Calculate the electric field, magnitude and direction, at point

P due to these charges. Use the coordinate system shown.

(b) Determine the electric potential at point P due to these

charges, using the usual choice for V = 0.

1 2Q = +4.72 × 10 C Q = -3.25 × 10 C-6 -6

3Q = +2.75 × 10 C a = 1.25 cm-6

101. Two conducting spheres are connected with a long wire. A total charge

of 1.65 × 10 C is placed on them. The radius of sphere 1 is 6.00 cm,-6

and the radius of sphere 2 is 2.00 cm.

(a) What is the charge on each sphere?

(b) What is the potential of each sphere, using the usual choice for V = 0.

(c) Find the electric field at the surface of each sphere.

o102. A non-conducting sphere, of radius R , has a positive charge

odistribution give by D(r) = Br for 0 < r # R , and D = 0 everywhere3

else. B is a constant.

(a) Calculate the total charge on the sphere.

(b) Calculate the electric field, magnitude and direction, at point P,

owhich is on the x-axis, a distance 3R from the center of the

sphere.

2 o(c) Calculate the electric field at a point P inside the sphere a distance of R /2 from its center (magnitude and

direction).

103. (a) Calculate the electric force (in Newtons) between two electrons a distance 3.00 × 10 m apart. -11

(b) Assume a proton has a radius of 1.00 × 10 m (about right). It has a positive charge equal in magnitude-15

to that of the electron. Calculate the charge density, assumed to be uniform.

(c) Given a potential function V = B x y z . Calculate the y-component of the electric field at the point2 4 3

x = 2.00, y = 3.00, z = 1.00.

(d) Calculate the term in x using the binomial expansion for the expression (1 - x ) . 6 2 3/2

(e) For the arrangement shown, what is the potential difference V(B) - V(A)?

104. Three point charges are along the x-axis a distance b (m) apart (see drawing).

(a) Calculate the x component of the electric field at point P, a distance b on the y-

1axis above Q .

(b) Calculate the electric potential, V, at point P using the usual choice for V = 0.

1 2 3Q = 1.00 :C; Q = 2.00 :C; Q = 3.00:C; b = 1.25 m

o105. A point charge, +Q, is placed inside a thin, hollow conducting sphere of radius R . Another

ohollow sphere, a non-conductor and radius 2R , surrounds the conducting sphere and has

the same center. The total charge on the non-conducting sphere is !2Q, and is uniformly

distributed.

o(a) Calculate the electric field, magnitude and direction, at a point R = 3/2 R .

o(b) Calculate the electric field, magnitude and direction, at a point R = 5R .

o106. A non-conducting sphere has a radius R . The positive charge distribution on this sphere can be

described by D(r) = B r , where B is a constant.3/2

(a) Calculate the total charge on the system.

o(b) Calculate the magnitude of the electric field at a point a distance 2R from the center of the

sphere.

o(c) Calculate the magnitude of the electric field inside the sphere at a point R /3 from the center of

the sphere.

(d) In part (c) is the electric field directed towards or away from the center of the sphere?

107. (a) Calculate the magnitude of the electric force between two protons (charge = +e) a distance

4.00 × 10 m apart. ________________________________________________________!13

(b) Use the binomial expansion to calculate the coefficient of the x term for the expression (1 ! x) . ______3 !7/3

(c) For the arrangement shown, what is the potential

A Bdifference V ! V ? _______________________

(d) A very long thin wire has a total charge of Q = +3.75 × 10 C. Its total length is 57.0 m. Calculate the!6

magnitude of the electric field a distance 3.72 mm away from the center of the wire, nowhere near either

end. __________________________________________________________________________

(e) An electron is accelerated from rest through a potential difference of 137 volts. Calculate the velocity of

the electron. ___________________________________________________________________

108. Three point charges are along the x-axis a distance b apart, as shown in

3the drawing. Point A is b from Q along the x-axis, and P is directly

above A at a distance b.

(a) Calculate the x-component of the electric field at point P.

(Numerical answer.)

(b) Calculate the electric potential (in volts) at point P due to the three charges.

1 2 3Q = +4.50 :C; Q = !2.75 :C; Q = !1.22 :C; b = 7.00 × 10 m!3

109. Two conducting spheres are connected with a long wire. Sphere 1 has

a radius of R = 17.0 cm and sphere 2 has a radius of r = 3.50 cm. A

charge of 60.0 pC is placed on the system.

(a) What is the charge in picocoulombs on sphere 1?

(b) Find the potential of sphere 1 in volts.

(c) What is the value of the electric field a distance 2r from the

center of sphere 2. Sphere 1 is very far away.

110. A non-conducting sphere has a negative charge distribution that can be described by

D = Br , where B is a constant.7/2

(a) Calculate the total charge on the system.

o(b) Calculate the magnitude of the electric field a distance 3R from the center of

the sphere.

o(c) Calculate the magnitude of the electric field at a point R /4 from the center of

the sphere.

(d) In part (c) is the electric field directed inward or outward?

o o o111. A spherical distribution of charge is modeled as D = Ar for r < R and D = 0 for r > R , where R is radius of the3

sphere.

o(a) If the total charge on the sphere is 5.67 × 10 C, and R = 0.027 m, find a numerical value for A.-6

(b) Calculate the magnitude of the potential at r = 0.0230 m. This should be a numerical value with units.

(Note: It is necessary to calculate E first.)

(c) If the sphere is positively charged, and v = 0 is chosen at the center of the sphere, give the sign of the

potential in (b), with a CLEAR explanation of how you got this sign.

o112. A spherical distribution of positive charge has a charge density given by D(R) = D R for3

o oR < R where D is a constant.

(a) Calculate the electric field at an arbitrary interior point of radius R.

(b) Find the magnitude of the potential difference between the center and the surface of

the sphere.

(c) If the total charge on the sphere is 1.60 × 10 C, and the radius of the sphere is 0.875!6

m, find the potential difference in (b) in volts.

1 4113. Charges Q !Q are at the corners of a square with sides a. Numerical values are given

below.

(a) Calculate a numerical value for the electric potential at point A, exactly in the center

of the square.

(b) Find a numerical value for the electric potential at point B, exactly in the middle of

one side of the square.

1 2 3Q = +4.25 :C; Q = !3.75 :C; Q = !6.75 :C;

4Q = +1.35 :C; a = 1.65 cm

o114. A spherical non-conductor of radius R has a positive charge distribution given by D =

oBR for R < R where D = 0 everywhere else. B is a constant.4

(a) Calculate the electric field at an arbitrary interior point a distance R from the

center of the sphere.

(b) Find the magnitude of the potential difference between the interior point at a

1 oradius R and the surface at R .

1(c) Calculate the energy stored in the electric field between R = 0 and R = R .

o115. A sphere of non-conductor of radius R is uniformly negatively charged with a total

charge - Q.

(a) Calculate the magnitude of the electric potential difference between the point

o oR /2 and R .

o o(b) Obtain the sign of V(R ) - V(R /2), and clearly explain how you got it.

o(c) With the usual choice of V = 0, calculate the potential at R /2.

Data: Use these constants (where it states, for example, 1 ft, the 1 is exact for significant figure purposes).

1 ft = 12 in (exact)1 m = 3.28 ft1 mile = 5280 ft (exact)1 hour = 3600 sec = 60 min (exact)1 day = 24 hr (exact)

earthg = 9.80 m/s = 32.2 ft/s2 2

moong = 1.67 m/s = 5.48 ft/s2 2

1 year = 365.25 days1 kg = 0.0685 slug1 N = 0.225 pound1 horsepower = 550 ft@pounds/s (exact)

earthM = 5.98 × 10 kg24

earthR = 6.38 × 10 km3

sunM = 1.99 × 10 kg30

sunR = 6.96 × 10 m8

moonM = 7.35 × 10 kg22

moonR = 1.74 × 10 km3

G = 6.67 × 10 N@m /kg-11 2 2

k = 9.00 × 10 N@m /C9 2 2

og = 8.85 × 10 F/m-12

electron chargee = -1.60 × 10 C-19

electronm = 9.11 × 10 kg-31

Note: Any integral from Table B-5 might be added to his list.