VDU_Physics_3_001

## Solved University-Physics Problems

Physics 3:  Introduction to Electricity & Magnetism

## <<<  Coulomb's Law  >>>

Javier Montenegro Joo

## jmj@VirtualDynamics.Org

<<<  Coulomb's Law  >>>

(1)   In the figure, two positive and equal electric charges, Q1 = Q2, interact with a third charge Q. Determine the resultant force on Q.

Solution.-

The sketch shows the forces that Q1 and Q2  exert on Q. These two forces have vertical and horizontal components, but their vertical components cancel one to the other and, the resultant force on Q is the sum of their horizontal components.

(2)  Electric charges Q1, Q2 and Q3 whose magnitudes are indicated in the figure, are placed at the vertexes of a right triangle.  Determine the force experienced by Q3 (vector, magnitude and direction).

Solution.-

The sketch shows the forces that Q1 and Q2  exert on Q3. The resultant force on Q3 is the sum of the forces that each charge makes on Q3:

(3) An electric charge   Q1 = 15.0 uC  is located at x = 2 m, another charge  Q2 = 6.0 uC  is placed at  x = 0.  Find the position –between these two charges- of a third negative charge Q3, so that this charge experiences no force at all.

Solution.-

Since Q3 placed between Q1 and Q2 does not experience any force, the forces that the first two charges make on the third one must be equal.

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(4) Positive electric charges  Qo, 2Qo, 3Qo y 4Qo    are placed each at the vertexes of a square of side b. Determine the resultant force on a positive charge Qo  placed at the center of the square.

Solution.-

The total force on a charge Qo placed at the center of the square is the resultant of the four forces acting on this charge.  First, compute the distance from each vertex of the square to its center; next, compute the four forces:

(5) In the figure, two cork small balls of mass m and electric charge Q, are hanging at the extremes of massless ropes of length L, which are separated a distance L. Due to mutual electric repulsion both balls try to go away and so each rope makes an angle ‘A’ with the vertical. Determine the weight of each ball.

Solution.-

Consider one of the balls: the angle ‘A’  between the rope and the vertical, results from the combined effect of the electric repulsion force  F  and the weight  mg  of the ball:

(6) Electric charges  Q1, Q2, Q3 y Q4  have been placed each at the vertexes of a square of side b.  It is known that  Q1 = Q2 = Q3 = Qo .      Determine the magnitude of  Q4,  so that  Q1  remains at equilibrium (without experimenting any force).

Solution.-

In equilibrium, the force exerted by Q4 on Q1 must equilibrate the forces of Q2 and Q3 on Q1.

(7) Two point charges Q1  and Q2   are placed on a line at  45o   with the horizontal x-axis and separated a distance L.   A third charge   Q  is placed on the line passing by  Q1  and Q2 , and at a distance b of Q.  Determine the relationship between Qand Q2    so that these two charges do not experience any force.

Solution.-

The 45° orientation of the line where the electric charges are placed is irrelevant.  Since Q1   and Qmust not experience any force:

(8) Two  equal small cork balls of mass m, are hanging from the same point by insignificant weight threads of length L. The balls are loaded each with a charge Qo, so that both separate a distance  d  due to mutual repulsion. Determine the charge Qo in each ball.

Solution.-

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(9) Five point electric charges of equal magnitude Qo are placed equidistantly on a semi-circumference of radius R, which is centered at the origin of coordinates. Determine the total force on a charge of magnitude Q placed at the origin of coordinates.

Solution.-

For didactic reasons the charges on the edge of the semi-circle (see the sketch below) have been numbered 1,2,…,5.  Since the charges on the edge of the semi-circle are all equal, and all of them are at the same distance to the center, then all of them exert the same strength on charge Q.

It is obvious in the sketch that the third charge exerts a horizontal force F3 on charge Q at the center.

It can be seen that the force between charges 1 and Q points downwards, while the force between charges 5 and Q, points upwards, therefore these two forces cancel each other.

The forces exerted by charges 2 and 4 on charge Q at the center, point both at 45o  with the x-axis, both have vertical and horizontal components, but it is evident that their vertical components cancel each other, remaining only their horizontal components along the x-axis.

(10) Consider a right triangle whose equal sides have a length b and, which has a charge Q in its right angle. Charges Qo and –Qo  have been placed respectively, in the two other vertexes of the triangle. Determine the total electric force on charge Q.

Solution.-

The sketch below shows the positive charge Q in the right angle of the right triangle and the charges –Qo and +Qo at its vertexes named 1 and 2, respectively.

It can be seen the attractive force F1 and the repulsive force F2 that charges -Qo and Qo exert on Q, respectively.   As both charges have the same magnitude, their vertical components F1y  and F2y  cancel each other. It can be seen that both charges, –Qo and +Qo generate a resulting force  F1x + F2x  in the –X  direction. Under these conditions: