Physics Help >> Index to 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.

E_l = F/q

Therefore, F = E_lq

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.

E_l = kQ/r²

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

For parallel plates, E_l = 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.

PROBLEMS

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 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 create 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. Air breaks down (loses its insulating quality) and sparking results if the field strength is increased to about 3.0*10⁶ N/C. (a) What acceleration does an electron experience in such a field? (b) if the electron starts from rest, in what distance does it acquire a speed equal to 10% of the speed of light?

15. Four point charges, each of magnitude 2.34*10¹ C, are placed at the corners of a square 40.8 cm on a side. If three of the charges are positive and one is negative, find the magnitude of the force experienced by the negative charge.

16. A large electroscope is made with "leaves" that are 78-cm-long wires with tiny 24-g spheres at the ends. When charged, nearly all the charge resides on the spheres. If the wires each make a 30 degree angle with the vertical, what total charge Q must have been applied to the electroscope? Ignore the mass of the wires.

17. Two point charges have a total charge of 560 uC. 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. How do I 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. What are the distribution of charges?

20. Calculate the electric field at the center of a square 52.5 cm on a side if one corner is occupied by a +45.0 x 10^-6 C charge and the other three are occupied by -27.0 x 10^-6 C charges?

21. Some walls in the human body have a double layer of surface charge with a layer of negative charge inside and a layer of positive charge of equal magnitude on the outside. Consider a model for such a cell in which the surface charge densities are +/- 0.5*10^-3 C/m² and the cell wall is 5.0*10^-9 m thick. Treat the cell wall as a parallel plate apparatus and find: a. the electric field magnitude in the wall between the two charge layers b. the potential difference between the inside and outside the cell. c. Which is the higher potential?

22. Three equal charges, each of +4.6 μC, are spaced along a straight line. Charge A, is at one end of the line of 1.8 m from the central charge, B. Charge C is the other side of charge B a distance of 2.2 m from charge B. What is the magnitude and direction of the total force on each charge?

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

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

25. Find the electric field at a point midway between two charges of +.000000030 C and +.000000060 C separated by a distance of 30.0 cm.

26. A positive charge of +5.00 μC is placed at the origin. A negative charge of -2.00 μC is placed 20.00 cm away, somewhere in the first quadrant. A third charge of +5.00 μC is placed in the first quadrant, 8.75 cm away from the negative charge. The angle, going from the original positive charge to the negative charge to the other positive charge, is 1.396 radians. Find the resultant force exerted on the third charge by the other two.

27. Two parallel plates 2.1mm apart, are charged so that the potential difference between the  plates is 36V. (i) what is the electric field strength between the plates (ii) sketch the lines of electric flux between the plates, showing the direction of the field. (iii) suggest three ways in which the system could be modified to increase its capacitance. (iv) a small particle charge of +180nC is placed midway between the plates. find the force on the particle due to the electric field, and the energy required to move the particle 0.7mm towards the positively charged plate.

28. How do you calculate the magnitude and direction of the acceleration of a particle given  the electrical field intensity?

29.Two equal, positive charges, q = 2.0 μC are located on the x-axis one at +0.3 m and the other at -0.3 m. A third charge Q = +4.0 μC is located on the y-axis at +0.4m. Find

a. the magnitude and direction of the resultant(net) force on Q

b. the electric field at the point (0.0,-0.4m)?

c. the potential at the point (0.0,-0.4m)?

30. The Moon and Earth are bound together by gravity. If instead the force of attraction were the result of each having a charge of the same magnitude but opposite in sign, find the quantity of charge that would have to be placed on each to produce the required force.

31. A clock face has negative point charges, -1q,-2q,-3q......,-12q, fixed at the positions of the corresponding numerals. The clock hands do not perturb the net field due to the point charges. If the hour hand points in the direction of the net electric field at the center of the clock, show that this corresponds to a time of 9:30.

32.

Two point charges are called an electric dipole. Show that the electric field at a distant point along the x axis is given by

E_x = 4k_eqa/x

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

34. A piece of Styrofoam having a mass m carries a net charge of –q and floats above the center of a very large horizontal sheet of plastic that has a uniform charge density on its surface. Apply Gauss' Theorem to determine the charge per unit area on the plastic sheet.

ANSWERS
1.

For solutions to all the problems on this page click here.

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.

For solutions to all the problems on this page click here.