AP Physics 2 Assignment – Electrostatics

Reading   Sections:  18.1 – 8; 19.1 – 4, Open Stax College Physics
                Sections:  16.1 – 9; 17.1 – 5, Giancoli
                Sections:  17.1 – 7; 18.1 – 6, Etkina et. al.

Objectives/HW

	The student will be able to:	HW:
1	Relate electrical phenomena to the motion and position of the fundamental charge found on electrons and protons and recognize the coulomb as the SI unit of charge and e as the elementary quantum of charge.	1 – 5
2	State and apply Coulomb’s Law to solve problems relating force, charge, and distance.	6 – 13
3	Define and apply the concept of an electric field and sketch field lines for a given distribution of charge and solve for the electric field strength at any point relative to a collection of point charges.	14 – 22
4	Define electric potential and potential difference and the volt and solve problems relating electric potential to charge, work or energy, electric field strength and distance.	23 – 32
5	Define and calculate potential and isolines for common charge distributions and solve related problems.	33 – 38

Homework Problems

1. A person holds one end of a glass rod and rubs the other end with a piece of silk.  This causes the rubbed end of the glass to become positively charged.  (a) Employ the ideas of conservation of charge and subatomic particles to explain the charge attained by the silk in this process.  (b) The held end of the glass remains neutral – why doesn’t the positive charge spread to this end?  (c) The glass is laid down on a metal surface – predict what happens to the charge of each end – explain your reasoning.

2. Suppose you are walking across a floor and your body picks up electrons in the process.  And then as you reach for a metal door knob a spark occurs – ouch!  (a) Based on this observation is the human body a conductor or insulator?  Explain.  (b) Just before the spark occurs, charge is rearranged in the metal door knob by the process of induction and this promotes the spark to occur – explain how and why.

3. A positively charged rod is brought near a neutral pith ball that is hanging from an insulating string.  At first the pith ball is attracted to the rod, but as soon as the two objects touch, the pith ball is repulsed from the rod.  (a) Explain why the pith ball at first is attracted to the rod.  (b) Explain why the pith ball is repulsed from the rod after it is touched by the rod.

4. When lightning occurs it is usually negative charge from a cloud transported to Earth.  In this process the charge passes through the air, which is an insulator, and goes to the ground, which is a conductor.  If the Earth as a whole has a net charge of zero, explain what forces make the charge leave the cloud, pass through the air, and go to the ground. 

5. Millikan’s oil drop experiment is repeated by a student.  The student uses the apparatus to measure and determine the charge on various drops of oil and records these values:  3.14, 1.61, 1.55, 6.24, 4.75, and 6.28 × 10^–19 C.  (a) Determine the best value (i.e. the mean) to report for the fundamental charge e consistent with these results.  (b)  Assess the success of the experiment in terms of random and/or systematic error and in terms of precision, accuracy, deviation, etc.  

6. The 0.25 m diameter metal sphere atop a certain Van de Graaff generator attains a charge of –2.5 μC when in operation.  A second metallic sphere of the same diameter is connected by a wire to the grounded base of the generator and attains a charge of +1.1 μC.  The center of the positive sphere is 0.40 m to the left of the negative sphere.  (a) Use Coulomb’s Law to determine the electrostatic force on each sphere.  (b) Because the spheres are metallic conductors the actual force will be different from that calculated – explain.  (c) In light of conservation of charge how can the difference in charge amounts on the two spheres be explained?

7. An alpha particle is equivalent to the nucleus of a helium atom and consists of two protons, and two neutrons, each mass 1.7 × 10^–27 kg.  The approximate separation of these particles is 1.8 × 10^–15 m.  (a) Determine the magnitude of the electric force repelling the protons from one another.  (b) Determine the magnitude of the gravitational force attracting one proton to the other.  (c) If you determined the net gravitational force including the effect of the neutrons would it be enough to equal the amount of repulsion?  Explain.  (d) Based on these calculations it can be concluded there must be another force besides gravity holding the nucleus together – what is it?  Explain the logic in concluding its existence. 

8. Suppose the gravitational attraction between two identical chunks of graphite happens to exactly equal the electrostatic repulsion of the two objects.  (a) Determine the mass and approximate size of each chunk if each has a charge amount equal to 1 e.  (b) How many carbon atoms are in each chunk (only one of which has lost or gained an electron)?  

9. Two ions interact electrostatically with a force of 0.018 nN when separated by a distance of 5.0 nm.  As a result of this, one particle accelerates 3.6 × 10¹⁵ m/s², 0.0° and the other accelerates 2.7 × 10¹⁵ m/s², 180.0°.  Calculate the charge and mass of each particle and speculate what each one is made of – can you be certain?

10. Three charges are arranged in a line as shown in the diagram below.  (a) Determine the net electric force on the middle charge.  (b) Determine the net electric force on the left charge.

11. As in the previous problem three charges are arranged in a line in order q₁, q₂, q₃ – however, this time the charge amounts and signs are unknown.  A distance d separates q₁ and q₂.  The net electrostatic force, F, acting on q₁ is equal and opposite the net electrostatic force acting on q₃. (a) Derive and simplify an expression for the net electrostatic force acting on q₂.  (b) Derive and simplify an expression for the distance between q₂ and q₃.  (c) Repeat part (a) but with net electrostatic F in same direction, not opposite.

12. Four charges are arranged in a square with sides equal to 10.0 cm.  All four charges are equal to +2.5 nC.  Determine the magnitude of electric force acting on any one of these four charges due to the presence of the other three.

13. Make a sketch of charges located in the x-y plane at the given coordinates:  q₁ = +7.50 μC at (-50.0 cm, 0.00 cm); q₂ = -7.50 μC at (+50.0 cm, 0.00 cm); and q₃ = +2.10 nC at (0.00 cm, +30.0 cm).  Charges 1 and 2 are fixed in place and charge 3 is free to move.   Given the mass of q₃ is 2.50 grams determine its acceleration due to its electrical interaction with q₁ and q₂.

14. (a) Determine the electric field acting on charge q₃ in the previous problem.  (b) Suppose that charge q₃ is removed and replaced by a charge q₄ = –4.0 nC; what is the electrostatic force on this new charge?

15. A pith ball of mass 0.20 grams and charge –3.0 nC hangs from an insulating thread and is repelled by the charged metallic sphere of a Van de Graaff generator.  It is noted that the thread is tilted by 17° when the pith ball hangs at rest 0.30 m from the center of the sphere.  (a) Determine the electric field at that position.  (b) If pith ball is moved to a position 0.30 m directly above the center of the sphere what will be the tension in the thread?  (c) In an electric field of this strength the electric force would be as great as gravity with what charge on the ball?  

16. A certain pith ball has a charge of +4.0 nC and is located at the origin of a coordinate system.  Determine the electric field at the following points in the x-y plane:  (a) (0.00 cm, -30.0 cm) and (b) (+15.0 cm, 0.00 cm).  (c) Repeat both (a) and (b) for a charge of -4.0 μC located at the origin.

17. Air becomes conductive when the electric field strength exceeds 3000 kN/C.  (a) Determine the maximum amount of charge that can be held by a metal sphere 0.15 m in radius.  (At a certain amount of charge, the field strength at the surface will be just great enough to cause sparks to radiate outward from the sphere, preventing further buildup of charge.) 
(b) Determine the acceleration rate of an electron at the surface just as a spark occurs.

18. (a) Derive an expression for the maximum amount of charge Q that can be “stored” on a sphere of radius R in terms of the maximum field strength E allowed by the surrounding air (beyond which dielectric breakdown occurs and sparks occur).  Express your result in terms of the permittivity constant.  (b) Based on this result by what factor must R increase to allow storage of charge ten times greater? one hundred times? one thousand times? 

19. In a demonstration with a Van de Graaff generator two spheres are given opposite charges of +2.0 μC and -2.0 μC.  Each sphere has a diameter of 30.0 cm and the centers of the spheres are 1.50 m apart.  (a) Make a sketch of the electric field surrounding the two spheres.  For the next two questions assume the two spheres produce fields equivalent to those produced by point sources (which is not exactly true).  (b) Determine the electric field at a point exactly half way between the two spheres.  (c) Determine the electric field at a point along a line connecting the centers and adjacent to the surface of the negative sphere.

20. (a) Make a careful sketch of the electric field lines around the charges in problem #10. 
(b) Make a careful sketch of the electric field lines around the charges in problem #12. 

21. Three charged particles are arranged initially at vertices of a 30°-60° right triangle.  When released from rest the two particles at each end of the hypotenuse accelerate initially in directions parallel to the legs of the triangle.  Ignore gravity.  (a) Given the particle at the right angle vertex has charge q, derive expressions for the charge amounts on the other two particles.  (b) Determine the direction of the initial acceleration of the particle with charge q.

22. A particle of charge q is located at the origin of a coordinate system.  Another charged particle is located on the x-axis at coordinates (d, 0).   The net electric field at the coordinates (–d, 0) is zero.  (a) Determine the charge on the second particle in terms of q.  (b) At a certain position on the y-axis the vector diagram showing the net electric field is an equilateral triangle.  Determine the electric field at this position on the y-axis.

23. How much energy does a single electron gain when passing through a voltaic cell with a potential difference of 1.5 volts?

24. A certain cell phone battery maintains a potential difference of 3.8 V and stores 40 kJ of energy when fully charged.  (a) Determine the total amount of charge that can pass through the battery before it “dies”.  (b) This quantity of charge is equivalent to what number of milliampere-hours (mA h)?  This value is usually shown on the label of a rechargeable battery.  Hint:  1 coulomb = 1 ampere * 1 second.

25. An “electron gun” accelerates electrons between two oppositely charged electrodes held at a certain potential difference.  Scientists planning a certain experiment need electrons to be “launched” at a speed of 3.2 × 10⁷ m/s.  (a) Determine the voltage requirement of the electron gun.  (b) What if the experiment requires protons launched at the same speed?  (c) Explain why the separation distance of the electrodes is not a factor in this calculation.  What factors might affect the distance used in the design of an electron gun?

26. A force of 0.053 N is required to move a charge of 37 mC a distance of 25 cm in an electric field.  (a) Determine the electric field strength, assuming it is constant.  (b) What is the amount of potential difference between the two points?

27. The earth is surrounded by an electric field of 150 N/C pointing toward its center.  This field can be assumed to be uniform and pointing downward within a classroom.  (a) What electric charge would have to be placed on a 0.10 gram pith ball in order for it to “levitate” – i.e. float in air without moving?  (b) Determine the potential difference between the ceiling and floor, which are separated by 3.0 m. 

28. Consider the information from the previous problem.  Suppose a 0.10 gram pith ball with electric charge -8.0 nC is dropped from the ceiling to the floor.  (a) Determine the work done by the electric field.  (b) Determine the difference in the impact speed of the ball that results from the presence of the electric field – in other words compare the impact speed caused by only gravity to the impact speed caused by both gravitation and electric force.

29. Spacecraft sometimes use ion propulsion to maneuver.  Suppose xenon ions (131.3 u, +1 e) are accelerated at 4.0 × 10¹¹ m/s² to a speed of 29000 m/s between two parallel oppositely charged grids and then ejected into space.  Assume a uniform electric field between the grids.  (a) Determine the electric field strength between the grids.  (b) Determine the potential difference between the grids.  (c) Determine the separation distance of the grids.

30. Two parallel metal plates are connected to a voltage source so that their potential difference is maintained at 350 V.  The two plates are separated by 1.00 cm.  An electron is released from rest at the negative plate and accelerates toward the positive plate.  (a) Determine the electric field strength between the two plates.  (b) Determine the amount of kinetic energy gained by the electron.  (c) Determine the speed of the electron just as it reaches the positive plate.

31. A beam of electrons with speed 8.0 × 10⁷ m/s enters the space between two oppositely charged parallel square plates moving perpendicular to the electric field and parallel to one side of the squares.  The plates are 3.00 cm on each side and separated by 0.500 cm.  The potential difference of the plates is maintained at 1300 volts.  (a) By what amount is the direction of the beam changed by the passage through the plates?  (b) What is the minimum speed of electrons that the beam can pass through without hitting one of the plates?

32. Could electric force cause an “orbit” instead of gravity?  Suppose a particle of mass 1.5 grams has charge 7.0 nC and “orbits” in a circular path of radius 0.30 m with period 2.0 s.  Ignore gravity.  (a) What strength electric field would be necessary to cause this motion?  The field’s direction would have what properties?  (b) An object with what charge could create such a field?  Is such a charge “doable”?  (c) Could this type of force interaction have a significant effect on an actual orbit of a celestial body like a planet or moon? 

33. (a) Calculate the electric potential relative to infinity for the locations and charges described in problem #16 above.  (b) Make a sketch of each point charge showing electric field lines and potential isolines.  (c) At any point in either sketch (or at any point in the universe!) the electric field points towards isolines of higher or lower potential?  Explain.

34. A sphere of radius 0.10 m has charge 1.5 μC and is fixed atop an insulating stand.  A pith ball of mass 0.13 grams and charge –1.2 nC is dropped from a location 0.50 m directly above the center of the sphere.  (a) Determine the electric potential at the release point and the point of impact with the sphere.  (b) Determine the potential difference of the impact point relative to the release.  (c) Use conservation of energy to determine the impact speed.  

35. Suppose the situation in the previous problem is repeated in all regards except the pith ball is positively charged.  (a) The pith ball would in fact fall downward after its release, though this is not obvious – how can it be determined which way the ball moves initially?  Explain.  (b) In spite of falling toward the sphere, the positive pith ball would not impact the sphere.  Explain how this can be ascertained.  (c) What would govern the position at which the ball would reverse direction and then fly away from the sphere?  Explain.

36. A point-like particle of mass m and charge q is released from rest and interacts with a stationary spherical charge Q with radius R.  Derive an expression for the maximum speed that can be attained by the particle in such an interaction.  Express your result in terms of the given quantities and the permittivity of free space.  Where must the particle start and where will it be when it attains the maximum speed?  Ignore all other forces.

37. Consider the charges described in problem #22 above.  (a) Determine the electric potential at the position where the electric field is zero.  (b) Determine the position on the x-axis at which the electric potential is zero.  (c) Is there any position at which both the field and the potential are zero?  Explain.

38. An equation for finding potential associated with gravity would be V_G = –GM/r.  (a) Use this equation to calculate potential at the surface of the Earth and at an altitude of 9.0 km (ignore significant digits for this one).  (b) Find the potential difference.  (c) Use the potential difference to calculate the increase in energy of a 100.0 kg person that rises to that altitude while flying in an airplane.  (d) Approximate the same value using U = mgh and compare the results.  Why is this technically only an approximation (although it is off by only 0.2%)?

Numerical Answers

3.2

10

32

9.33 × 10¹⁶

12°

5.0 × 10⁻²⁷ kg (3.0 u) and 6.7 × 10⁻²⁷ kg  (4.0 u)

1.86 × 10^–9 kg

1.57 × 10⁻¹⁹ C

1.6 × 10⁻¹⁹ C (1 e) and 3.2 × 10⁻¹⁹ C (2 e)

±9.8 nC

–6.3 μC

-6.5 μC

7.5 μC

11 kC

2900 mA h

6.0 × 10⁻³⁵ N

1.1 × 10⁻⁵ N away from center

2.9 × 10⁻⁵ N, 180.0°

5.1 × 10⁻⁵ N, 0.0°

1.4 × 10⁻³ N

1.4 mN, 180.0°

0.15 N, left

0.15 N, right

71 N

400 N/C, 270°

1400 N/C

1600 N/C, 0°

35 kN/C

64 kN/C toward the neg. sphere

2.0 × 10⁵ N/C toward the sphere

340 kN/C, 0.0°

400 kN/C, 90°

540 kN/C

630 kN/C pointing radially inward

810 kN/C toward the neg. sphere

1600 kN/C, 180°

1.1 mm

5 × 10⁻³ m/s slower

3.1 m/s

1.1 × 10⁷ m/s

6.4 × 10⁷ m/s

0.286 m/s², 0.0°

5.3 × 10¹⁷ m/s²

2.4 × 10^–19 J (or 1.5 eV)

5.6 × 10⁻¹⁷ J

-3.6 μJ

8.81 MJ

8.82 MJ

+88.1 kJ/kg

– 62.4035 MJ/kg

– 62.4916 MJ/kg

120 V

240 V

360 V

450 V

570 V

2.9 kV

27 kV

110 kV

–120 kV

140 kV

–240 kV

5.3 MV