COULOMB'S LAW AND ELECTRIC FIELDS
BRIEF YOURSELF
One of nature's most spectacular display of electricity is the lightening observed during a thunder storm. Electricity originates from electric charges. All material bodies are known to possess electric charges. Electrons carry negative charges while protons carry positive charges in the nucleus of an atom. The unit of charge is Coulomb.
The force exerted by electric charge is described by the Coulomb's law of force. The force is attractive between unlike charges and repulsive between like charges. The force between two charges is directly proportional to the product of the two charges, and inversely proportional to the square of the distance between the two charges. The force on each charge is equal in magnitude but opposite in direction. For a pair of charges q1 and q2, separated by a distance r, the Coulomb's law may be stated as follows:
F = k(q1q2/r2) ....................................... (1)
The constant of proportionality, k = 8.99x109 N.m2/C2. Such a force is transmitted by the presence of an electric field. The electric field E due to a point charge q is,
E = k(q/r2) .................................. (2)
Electric force and electric field are vectors. Hence, they have magnitudes and directions. The electric force F and electric field E are related as follows:
F=qE ......................................... (3)
Where the force is on charge q due to the presence of an electric field at the position of q.
Multiple charges and the Principle of Superposition
When more than two charges are present, the net force on any one charge is equal to the vector sum of each of the forces produced by other charges. For example, the force on charge q1 due to the presence of charges q2 and q3, is the superposition of the forces exerted by q2 and q3. That is, the net force F on charge q1 is,
F = F12 + F13 ........................................... (4)
where, F12 is the force on q1 due to the presence of charge q2 and F13 is the force on q1 due to charge q3. While solving a problem, it is useful to rewrite equation (4) in its component form as follows:
Fx = (F12)x + (F13)x
Fy = (F12)y + (F13)y
Principle of superposition also applies to the electric fields produced by multiple charges. That is, the net electric field at a point due to several charges is the vector sum of the electric fields due to individual charges.
TEST YOURSELF
1. Find the approximate number of unit charges present in 1 C of electric charge.
(1 unit of charge = 1.6x1019 C)
2. The mass of an electron is 9.11x103 kg. Electron carries a single unit of charge equal to 1.605x1019 C. Calculate the ratio of the electric force and the force of gravity between two electrons.
3. The nucleus of a carbon atom has 12 units of positive charges and the nucleus of a neon atom has 22 units of positive charges. Find the force between the two nuclei if they are 5x109 m apart.
4. (a) Find the magnitude and direction of the electric field at a distance of 0.45 m from a point charge +5 mC. (b) Find the magnitude and direction of the force on a second point charge +2 mC placed at 0.45 m from the first point charge. [1 mC=1x106 C]
5. Two point charges +6 nC and 8 nC are 0.5 m apart. (a) Find the electric field at a point between the two charges, at a distance of 0.25 m from the 8 nC point charge.
(b) Find the net force on a third point charge +4 nC, placed at the same point as in (a).
[1 nC=1x109 C]
6. Two point charges +7 mC at x=0 and 9 mC at x=0.75 m are placed along xaxis. (a) Find the resultant electric field at x=2.5 m. (b) What is the net force on a point charge 3 mC placed at x=2.5 m?
7. A positive point charge of amount +25 mC experiences an attractive force of 0.012 N if placed at a distance of 0.5 m from a second point charge. (a) Find the magnitude and sign of the unknown charge. (b) Find the magnitude and direction of the electric field at 0.5 m due to the unknown charge.
8. Two point charges +5.0 mC at x=0 and +7.0 mC at x=1 m are placed along xaxis. (a) Where should a third point charge +3 mC be placed so that it experiences no electric force? (b) Where should a third point charge of amount 4 mC be placed so that it experiences no electric force? (c) Does the answer depend on sign and magnitude of the third charge? (d) What is the electric field at the distance found in (a) and in (b).
9. Two point charges 4nC at x=0 and +2nC at x=1 m are placed along xaxis as shown below. Find the distance where a third point charge 2 nC may be placed with no net force on it.
10. Two point charges +2 mC at x=1 m and +2 mC at x=1 m are placed along
xaxis as shown below. (a) Find the magnitude and direction of the electric field at
y=3 m. (b) Find the magnitude and direction of the net force on a third charge +3 mC placed at y=3 m.
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HINT: Freebody diagram for problem #10..
11. Three point charges are placed in the following way: q1=+0.05 mC at x=0, y=0;
q2=0.07 mC at x=3 m, y=0; and q3=+0.04 mC at x=0, y =5 m. as shown below. (a) Find the magnitude and direction of the electric field at y=3 m. (b) Find the resultant force on +0.02 mC point charge at y=3 m.
12. Four equal point charges of amount +5 mC are placed at the four corners of a square as shown below. Find the magnitude of the net force on each charge due to three other charges.
13. Electric field of magnitude 1000 N/C is directed horizontally along positive xaxis in a region of empty space. A point charge of +5 mC and negligible mass is released from rest at x =1.0 m. How long will it take for this charge to reach x=10 m?
14. The electric field near the surface of the earth at a certain place is 140 N/C directed vertically downward and the acceleration due to gravity is 9.8 m/s2. How much charge must a coin of mass 0.015 kg possess so that it experiences no net force?
15. In a TV set, an electron moving horizontally at a speed of 1x108 m/s encounters an electric field of magnitude500 N/C directed vertically upward for 1.5x106 s. What is the displacement of the electron due to the electric field?
16. A small charged ball of mass 0.5x104 kg hangs vertically from a thread at rest. An electric field of 4500 N/C is directed vertically upward. What is the charge on the ball if the tension in the string is
(a) zero
(b) 3x103 N.
17. A tiny charged ball of mass 0.75x103 kg, tied at the end of a thread interacts with a horizontal electric field of 850 N/C. The ball is in equilibrium as shown in the Figure: below. The angle made by the thread with the vertical is 30o. Determine the sign and the magnitude of the charge.
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HINT: Freebody diagram for problem #17
18. An electron is projected horizontally with a forward speed of 2x107 m/s. The electron goes 0.5 m before momentarily coming to rest and retracing its path due to the presence of electric field. What is the magnitude and direction of the field?
19. A certain molecule contains two opposite charges of amount 5x1019 C each and, separated by a distance of 3.5x1010 m (see diagram below). Find the magnitude of the electric field at a distance of 5x107 m from the center of the two charges (a) along the line of two charges and (b) normal to the line of two charges passing through the center.
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HINT: Free body diagrams for problem #19
(a)
(b)
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20. An electron has an initial speed of 4x106 m/s along xaxis. At x=0, it encounters a uniform electric field of 500 N/C along positive yaxis. (a) Find the acceleration of the electron and (b) displacement of the electron along yaxis after it travels 0.015 m along xaxis.
HINT
Free body diagram for problem #20 (b).
21. The orbit radius of the electron moving around the nucleus in a hydrogen atom is
5.29x1011 m. The centripetal force on the electron comes from the Coulomb's electric force as shown below. Find the speed of the electron.
22. Point charge 5nC is placed at each of the corners of a square, 0.15 m on the side. Find the (a) electric field at one corner due to presence of 3 charges at other corners and (b) the electric force on a charge at each corner.
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HINT: Free body diagram for problem #22(a)
23. Each of two small equally charged spheres of mass 0.01 kg is suspended from the same point by a silk fiber 0.4 m long. The repulsion between the two keeps them 0.2 m apart as shown. What is the charge on each sphere?
Diagram for problem #23
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HINT: Freebody diagram for problem #23
24. Two equally charged pith balls, each of mass 2x103 kg are hung from ends of a string 0.5 m long. The repulsive electric force keep them separated making an angle 45o at the vertex as shown below. What is the charge on each ball and tension in the string?
Diagram for problem #24
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HINT: Freebody diagram for problem #24
25. Two identical small metal spheres contain unequal unknown charges. The attractive force experienced by each charge is 2.00x104 N when separated by a distance of 0.15 m. After the two charges touch each other and then removed to the original distance, the repulsive force experienced by each sphere is 5.00x105 N. Find the two unknown initial charges.
CHECK YOURSELF: Solution of the problems avilable in the Crumb Library.
GRADE YOURSELF
Subject: Charge & Electric Fields
Topic 
Question Numbers 
Number Incorrect 

Charges 
1 

Coulomb's law 
2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15, 16, 17, 18, 20, 21, 23, 24, 25 

Principle of Superposition 
10, 11, 12, 19, 22 
