Ch 22: Electric Force & Field; Gauss' LawSee all chapters
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Ch 20: The First Law of Thermodynamics
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Ch 22: Electric Force & Field; Gauss' Law
Ch 23: Electric Potential
Ch 24: Capacitors & Dielectrics
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Coulomb's Law (Electric Force)

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Sections
Electric Charge
Charging Objects
Charging By Induction
Conservation of Charge
Coulomb's Law (Electric Force)
Electric Field
Parallel Plate Capacitors
Electric Field Lines
Dipole Moment
Electric Fields in Conductors
Electric Flux
Gauss' Law

Concept #1: Coulomb's Law

Practice: If the force between two charges is F when the distance is d, what will the force between the two charges be if they were moved to a distance of 2d

Example #1: Charges In A Line (Find Zero Force)

Practice: In which direction will the – 1 C charge move? If it has a mass of 10 g, what will its initial acceleration be?

Example #2: Charges In A Triangle (Rank Force Pairs)

Example #3: Charges in a Plane

Example #4: Exploiting Symmetry

Practice: What is the direction of the net force on the charge at the center of the square in the following figure?

Example #5: Electroscope (Find Charge)

Additional Problems
A point charge Q = -12 μC, and two other charges, q 1 and q2, are placed as shown. The electric force components on charge Q are Fx = +0.005 N and Fy = -0.003 N. Find charge q1 and q2.
Two small beads having charges q1 = +9Q and q2 = -Q are fixed on a horizontal insulating rod. The distance between q1 and q2 is d = 1.5 m  The third bead with charge q 3 = +Q is at the right hand side of the second bead and is free to slide along the axis of the rod. At what position x is the third bead in equilibrium?
Two charges q1 and q2 are separated by a distance d and exert a force F on each other. What is the new force F ′ , if charge 1 is increased to  q ′1 = 5 q1, charge 2 is decreased to q ′2 = q2 / 2, and the distance is decreased to  d ′ = d / 2? Choose one. 1. F ′ = 5/2 F 2. F ′ = 5/4 F 3. F ′ = 20 F 4. F ′ = 25/4 F 5. F ′ = 50 F 6. F ′ = 25/2 F 7. F ′ = 100 F 8. F ′ = 5 F 9. F ′ = 25 F 10. F ' = 10 F
Two charges, Q1 and Q2, are separated by a certain distance R. If the magnitudes of their charges are doubled and their separation is halved, then what happens to the electrical force between these charges? A) It increases by a factor of 16. B) It is quadrupled. C) It is doubled. D) It stays the same.
The force of attraction between a -40.0 μC and +108 μC charge is 4.00 N. What is the separation between these two charges? A) 1.13 m B) 3.67 m C) 3.12 m D) 2.49 m
Two point charges of +60.0 μC and -12.0 μC are separated by a distance of 20.0 cm. A +7.00 μC charge is placed midway between these two charges. What is the electric force acting on this charge because of the other two charges? A) 4.53 N directed towards the positive charge B) 453 N directed towards the negative charge C) 4.53 N directed towards the negative charge D) 453 N directed towards the positive charge
Two identical small charged spheres are a certain distance apart, and each one initially experiences an electrostatic force of magnitude F due to the other. With time, charge gradually leaks off of both spheres. When each of the spheres has lost half its initial charge, the magnitude of the electrostatic force will be (a) 1/2 F (b) 1/16 F (c) 1/8 F (d) 1/4 F
A positive charge Q is fixed on a very large horizontal frictionless tabletop. A second positive point charge q is released from rest near the stationary charge and is free to move. Which statement best describes the motion of q after it is released? (a) As it moves farther and farther from Q, its acceleration will keep increasing. (b) As it moves farther and farther from Q, its speed will decrease. (c) Its speed will be greatest just after it is released. (d) Its acceleration is zero just after it is released. (e) As it moves farther and farther from Q, its speed will keep increasing.
A positive test charge q is released near a positive fixed charge Q. As q moves away from Q, it will move with A) increasing acceleration. B) constant velocity. C) decreasing acceleration. D) constant acceleration.
Three charges, q1, q2 and q3 are placed left to rigth in a straight line. The distance between q1 and q2  is twice the distance between q 2 and q3. Charge q1 is negative. If the force on charge q3 is zero, which of the following statements about q 2 is true? A) q2 is negative with a magnitude twice that of q 1 B) q2 is negative with a magnitude half that of q 1 C) q2 is positive with a magnitude one fourth that of q 1 D) q2 is positive with a magnitude one ninth that of q 1 E) q2 is positive with a magnitude equal that of q 1
Three identical point charges, Q = 3μC, are placed at the vertices of an equilateral triangle as shown in the figure. The length of each side of the triangle is d = 0.15m. Determine the magnitude and direction of the total electrostatic force on the charge at the top of the triangle.
At what separation will two charges, each of magnitude 6.0 μC, exert a force of 0.70 N on each other? A) 1.1 x 10-5 m B) 0.23 m C) 0.48 m D) 0.68 m E) 1.4 m
Find the magnitude and direction of the net electrostatic force on q 1 due to q 2 and q3 as shown in the figure. The charges are q 1 = +4.0 μC, q2 = -6.0 μC, q3 = -5.0 μC.
Five charges are equally spaced along the x axis. Each charge has the same magnitude e, but some of the charges are +e and some are −e. Four different configurations of charge are labelled A, B, C, D in the figure. Rank the magnitude of the force on the middle charge for the different configurations with largest first and smallest last. (1) C,D,A,B (2) C,A,D,B (3) D,C,A,B (4) A,C,D,B (5) C,B,D,A
An oil droplet with 4 excess electrons is held stationary in a field of 1.27x10  4 N/C. What is the radius of the oil drop? (The density of the oil is 824 kg/m3, e = 1.60 x 10 -19 C.)
Three charges are placed at the following (x, y) coordinates: charge + 6.0x10  -6 C at (0, 0.5 m), charge + 2.0x10 -6 C at (0.7, 0 m), and charge - 4.0 x 10 -6 C at (0.7 m, 0.5 m). Calculate the electrical force on a point particle with charge -1.6 x 10-6 C at the origin (0,0).
What is the difference between the electric force on a charge q when moved from a distance d to a distance 3d. Consider the initial electric force to be F0.  
Two identical,  0.25 μC charges hang from the end of electroscope leaves. If, in equilibrium, the electroscope leaves deflect from one another to an angle of 40° from the horizontal axis, as shown in the figure, what is the mass of each charge?  
A dipole with negative charge of -5.0 x 10 -6 C on the x-axis at x = -3.0 cm, and a positive charge of 5.0 x 10-6 C on the x-axis at  x = + 3.0 cm. A negative point charge of  q = -4.0 x 10-6 C is placed on the y-axis at  y = -4.0 cm. a) Draw the two forces acted on the negative point charge  q = -4.0 x 10 -6 C (in arbitrary scale, but correct DIRECTION). b) How big is the TOTAL eletrostatic force acted on the -4.0 x 10 -6 C charge? What is its direction? (i.e., the angle with respect to the x-direction).
A positive test charge q is released near a positive fixed charge Q. As q moves away from Q, it will move with  A) constant velocity. B) constant acceleration. C) decreasing acceleration. D) increasing acceleration. 
Two equally charged spheres of mass 1.0 g are placed 2.0 cm apart. When released, they begin to accelerate at 779 m/s2. What is the magnitude of the charge on each sphere? A) 160 nC B) 130 nC C) 190 nC D) 100 nC
What is the magnitude of an electric field that balances the weight of a plastic sphere of mass 2.2 g that has been charged to -3.0 nC? A) 7.2x106 N/C B) 8.1x105 N/C C) 1.5x106 N/C D) 2.2x106 N/C
Two charged particles of equal magnitude (−Q and +Q) are fixed at opposite corners of a square that lies in a plane (see figure below). A test charge +q is placed at a third corner. What is the direction of the force on the test charge due to the two other charges?
Two charged particles of equal magnitude (−Q and +Q) are fixed at opposite corners of a square that lies in a plane (see figure below). A test charge +q is placed at a third corner. What is the direction of the force on the test charge due to the two other charges?If F is the magnitude of the force on the test charge due to only one of the other charges, what is the magnitude of the net force acting on the test charge due to both of these charges?
Two positive point charges Q and 2Q are separated by a distance R. If the charge Q experiences a force of magnitude F when the separation is R, what is the magnitude of the force on the charge 2Q when the separation is 2R? A) F/4 B) F/2 C) F D) 2F E) 4F
An equilateral triangle  ABC. A positive point charge +q is located at each of the three vertices A, B, and C. Each side of the triangle is of length  a. A point charge Q (that may be positive or negative) is placed at the mid-point between  B and C. Is it possible to choose the value of Q (that is non-zero) such that the force on Q is zero? A) Yes, because the forces on Q are vectors and three vectors can add to zero. B) No, because the forces on Q are vectors and three vectors can never add to zero. C) Yes, because the electric force at the mid-point between  B and C is zero whether a charge is placed there or not. D) No, because the forces on Q due to the charges at  B and C point in the same direction. E) No, because a fourth charge would be needed to cancel the force on  Q due to the charge at  A.
Electrical and gravitational forces follow similar equations with one main difference: A) Gravitational forces obey the inverse square law and electrical forces do not. B) Electrical forces obey the inverse square law and gravitational forces do not. C) Gravitational forces are always attractive but electrical forces can be attractive or repulsive. D) Electrical forces attract and gravitational forces repel.
Four charges are arranged as shown in the figure below each with a positive (red) charge or a negative (blue) charge. Find the net total force (magnitude and direction) exerted on the charge with + 6.0 micro coulomb by the other three charges.
Three point charges +3 μC, +4 μC, and -7 μC are placed along the x and y axis with the values and positions shown. What is the net force, magnitude and direction, acting on the +4 μC charge? (a) 198 N, 37° above +x axis (b) 298 N, 47° above -x axis (c) 342 N, 53° above +x axis (d) 305 N, 53° below +x axis
Two identical small charged spheres hang in equilibrium with equal masses as shown in the figure. The length of the strings are equal and the angle (shown in the figure) with the vertical is identical.The acceleration of gravity is 9.8 m/s 2 and the value of Coulomb’s constant is 8.98755×10 9 N m2/C2.
Four charges −5×10−9 C at (0m, 0m), −8×10−9 C at (1m, 5m), 9×10−9 C at (3m, 1m), and 7×10−9 C at (3m, −3m), are arranged in the ( x, y ) plane (as shown in the figure below, where the scale is in meters).Find the magnitude of the resulting force on the −5 nC charge at the origin [coordinates, (0m, 0m)].
Two identical spheres A and B carry charges Q and 2Q, respectively. They are separated by a distance much larger than their diameters. A third identical conducting sphere C is uncharged. Sphere C is first touched to A, then to B, and finally removed. As a result, the electrostatic force between A and B, which was originally F, becomes: (1) 5F/16 (2) F/4 (3) 3F/8 (4) F/8 (5) 7F/16
A charge q1 = -18.0 nC hangs from a string. It has a mass of 2.9 g. A second charge of q2 = 53.7 nC is brought to a distance of 2.3 cm from q 1. Find the angle θ assuming the system is at rest as shown in the figure. A. 15° B. 21.3° C. 30° D. 45° E. 47.3°
Three charges are positioned as indicated in the figure. What are the horizontal and vertical components of the net force exerted on the +15 µC charge by the +11 µC and + µC charges?
An alpha particle with charge +2.0 e is sent at high speed toward a silver nucleus with charge +47e. What is the electric force acting on the alpha particle when the alpha particle is 4.2 × 10−14 m from the silver nucleus? The value of the Coulomb constant is 8.99 × 10  9 N·m2/C2.1. Unable to determine2. 12.2639 N, repulsive3. 13.0467 N, repulsive4. 12.2639 N, attractive5. 13.0467 N, attractive6. None of these
What is the net electric force on the 1 C charge? Consider the 1 C to placed directly in the center of the arrangement.