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## Physics Homework Help

### Electric Charge and Electric Field

#### ELECTRIC CHARGE

Atoms and molecules have the same number of protons as electrons and are neutral (without overall charge).  Electrons can be transferred from one object to another.  When this happens, there is an excess of electrons in one place and a deficiency of electrons in another. Charge is the result of an excess or deficiency of electrons.  Where an object has excess electrons, the object is negatively charged. Where there is a deficiency of electrons, the object is positively charged. Electric charge is usually represented in equations by the letter q or Q.

#### ELECTROSTATIC

"Electrostatic" pertains to electric charges at rest or to fields or phenomena produced by stationary charge(s).

#### COULOMB'S LAW

Two charges, Q and q, separated by a distance, r, each experience a force of magnitude

F = k|Qq|/r2 where k = 9 x 109 and |Qq| is the positive value of the product of Q and q

Charges of the same sign repel, of different signs attract each other.

#### ELECTRIC FIELD

The charge , Q, causes an electric force on every other charge, q. Q is called the source charge as it is considered to be the cause of the electric field, while q is called a test charge. The field is a vector quantity. The direction of the field is by definition the direction of the force on a positively charged object: the field points away from a positive source charge, and toward a negative source charge.

A common definition for electric field is a region of space where a positive test charge experiences a force.

Electric field intensity (sometimes just called electric field) is the force per unit charge experienced by a point charge somewhere in space.

El = F/q

Therefore, F = Elq

That is, electric field is the force produced by a source charge, Q, exerted on every coulomb of charge of a test charge at a distance r away from the source of the field.
El = kQ/r2

For parallel plates, El = ΔV/d, where ΔV is the potential between plates, and d is the distance between the plates.

For parallel plates, El = σ/ε0 where σ is the surface charge density C / m2.

#### GAUSS'S THEOREM

The total flux, ε, through a closed surface is equal to 1/ε0 times the total charge contained within it. The location of the charge(es) within the sphere does not matter.

### Electric Charge and Electric Field Problems

(See bottom of page for answers.)

1. Starting with a neutral electroscope, show the charge distribution and action of the leaves when the electroscope is first touched by a positively charged object, and then by a large neutral object.

2. Starting with a neutral electroscope, show the charge distribution and action of the leaves when the electroscope is brought near a negatively charged object.

3.
a. Two objects are identical in every way except that one is neutral, and the other has 2 excess electrons. Show what happens to the distribution of charges when the two objects are brought into contact and then released.
b. If the charge on the electron is "-1" what is the charge on each of the two objects after they are separated?

4.
a. Two objects are identical in every way except that one is deficient by two electrons, and the other has 4 excess electrons. Show what happens to the distribution of charges when the two objects are brought into contact and then released.
b. If the charge on the electron is "-1" what is the charge on each of the two objects after they are separated?

5. Two neutral identical objects , A and B, are in contact and brought near a negatively charged object, C. While in the presence of C, A and B are separated. What are the relative charges of A and B? Draw diagrams to show the charge distributions at each step.

6. A charge of -2 x 10-6 C experiences a force of 0.08 N [left]. What is the electric field at that point?

7. A charge of +3.0 x 10-6 C is 0.25 m away from a charge of -6.0 x 10-6 C.
a. What is the force on the 3.0 x 10-6 C charge?
b. What is the force on the -6.0 x 10-6 C charge?

8. Three charges, q1 = 4 x 10-6 C, q2 = -2 x 10-6 C, and q3 = 5 x 10-6 C are placed at the corners of a square with>sides 0.30 m. What is the field at the fourth corner?

9. A charged droplet of mass 5.88 x 10-10 kg is hovering motionless between two parallel plates. The parallel plates have a potential difference of 24000 V and are 2.00 mm apart. What is the charge on the particle? By how many electrons is the particle deficient?

10. Four point charges form the vertices of a square with sides = L. Two diagonally opposite charges have a charge of 2.25 C each. The other two charges are identical to each other and each have a charge, q. If there is no net force on either of the 2.25 C points, what is the value of q?

11. Two point charges lie on the x-axis. A charge of 9.9 C is at the origin, and a charge of -5.1 C is at x=10cm.
a. At what position x would a third charge q3 be in equilibrium?
b. Does your answer to part a depend on whether q3 is positive or negative? Explain.

12. Two particles each with a positive charge of q are placed on the vertices of a square having sides a. A third particle with a positive charge Q is placed at the center of the square. What is the force on the particle at the center of the square?

13. A charge of 6.00*10-9 C and a charge of -3.00*10-9 C are separated by a distance of 60.0 cm. Find the position at which a third charge of 12.0*10-9 C can be placed so that the net electrostatic force on it is zero.

14. An electron enters a region where the field strength is 3.0*106 N/C. (a) What is the electron's acceleration? (b) Starting from rest, how far does the electron travel to acquire 10% of the speed of light?

15. Four point charges, each of magnitude 2.34*10¹ C, form a square with sides 40.8 cm. If three of the charges are positive and one is negative, find the magnitude of the force experienced by the negative charge.

16. Two 24-g spheres are each attached to the bottom of very light 78 cm wires. When the wires are joined at the top, they each form an angle of 30 degrees to the vertical. What is the total charge on the spheres?

17. Two point charges have a total charge of 560 μC. When placed 1.10m apart, the force each exerts on the other is 22.8N and is repulsive. What is the charge on each?

18. Explain how to calculate the amount of free charge in a wire.

19. A sphere with a charge of -50 is centered within a hollow sphere having a charge of -100. Describe the distribution of charges.

20. A square with sides 52.5 cm is formed by a +45.0 x 10-6 C charge at one corner and -27.0 x 10-6 C charges at each of the other corners. What is the electric field at the center?

21. A -3.5 x 10-10 C point charge is fixed near the Earth's surface. An electron is placed near the point charge so that the electric force acting on the electron cancels the electron’s weight. Where is the electron relative to the point charge?

22. Three point charges, each +4.6 μC, form a straight line. Charge A is 1.8 m from the central charge, B. Charge C is 2.2 m from charge B. What is the magnitude and direction of the net force on each charge?

23. An 4.50 μC electric charge is in an electric field with a y-component Ey = 4000 N/C, an x-component Ex = 700 N/C and a z-component Ez = 0. What are the magnitude and direction of force on the charge?

24.Two charges, +q and 4q, are 1 m apart. What are the location, magnitude and sign of a third charge, Q, placed so that the entire system is at equilibrium?

25. Find the electric field midway between charges of +.000000030 C and +.000000060 C 30.0 cm apart.

26. A -2.00 μC forms the apex of a triangle, while two +5.00 μC charges form the base.  One of the +5.00 μC charges is 20.00 cm from the apex, and the third charge is 8.75 cm away from the apex.  The angle at the apex is 1.396 rad.   Find the net force on the third charge.

27. Two parallel plates 2.1mm apart have a 36V potential difference. (i) What is the electric field strength between the plates? (ii) Sketch the electric flux lines between the plates, and show the direction of the field. (iii) Suggest three ways to increase capacitance. (iv) Find the force on a +180nC particle placed midway between the plates, and the energy required to move the particle 0.7mm towards the positively charged plate.

28. Explain how to calculate the magnitude and direction of the acceleration of a particle given  the electrical field intensity.

29. A +2.0 μC charge is located on the x-axis at +0.3 m and another at -0.3 m. A third charge, +4.0 μC, is located on the y-axis at +0.4m. Find
a. the net force on the third charge
b. the electric field at (0,-0.4m)
c. the potential at (0,-0.4m)

30. Suppose that equal and opposite charges were placed on the Earth and the Moon.  What amount of charge on each would supply an electrical force equal to the gravitational force between them?

31. Point charges, -1q,-2q,-3q......,-12q, are fixed at the corresponding positions on the face of a clock.  What is the direction of the electric field?

32. Two non-coincident point charges on the x axis are each separated by a distance, a, from the origin. Show that at a distant point along the x axis the electric field is given by

Ex = 4keqa/x

33. Find the total electric flux through a spherical shell placed in a uniform electric field.

34. A tiny plastic sphere (mass = m, charge = –q) hovers above a large horizontal plastic sheet having a uniform charge density on its surface. Use Gauss' Theorem to find the sheet’s charge per unit area.

35. An electron with an initial kinetic energy of 1.60 × 10–17 J decelerates to rest over a distance of 10.0 cm. What are the magnitude and direction of the electric field that stopped the electron?

### Answers to Electric Charge and Electric Field Problems

Selected solutions are printed below.

1.

31. All angles are measured relative to 12 o'clock at zero degrees. Assigning a value of "1" to the electric field at the center of the clock face due to the charge at the 1 o'clock position, the x and y components of the field are given by sin30º and cos30º; the x and y components of the field due to the charge at 2 o'clock are given by 2sin60º and 2cos60º; and so on. The net field subtends an angle at the center of the clock face given by
tan-1(-22.39/6.00) = -75º,
which corresponds to a time of 9:30.

qx-componenty-component
-1 0.50 0.87
-2 1.73 1.00
-3 3.00 0.00
-4 3.46 -2.00
-5 2.50 -4.33
-6 0.00 -6.00
-7 -3.50 -6.06
-8 -6.93 -4.00
-9 -9.00 0.00
-10 -8.66 5.00
-11 -5.50 9.53
-12 0.00 12.00

totals -22.39 6.00