August 1, 2021

# Solution Set 7

Physics 208-02 – Introductory Physics II
Solution Set 7
Nazareth College
Department of Chemistry & Biochemistry
Robert F Szalapski, PhD – Adjunct Lecturer
www.CallMeDrRob.com
Spring 2012

## § Chapter 16, Problem 13

Given an equilateral triangle with an charge at each vertex and with side lengths of . Compute the magnitude and direction of the force on each charge. See Figure 1.

First of all, notice that each pair of charges will exert equal and opposite charges on each aligned with the sides of the triangle as shown. This is a total of six forces, and all will have the same magnitude due to the same charges at the same separation, but directions are different. Note also that each angle of the triangle is , and hence the angle between the pair of forces at each vertex will also have an angle of .

We really want to take advantage of symmetry, and the easiest force to consider if the net force at the top of triangle as shown in Figure 2. Let’s compute the magnitude of each force, and then let’s compute the magnitude of the and components as indicated.

Notice that the component is opposite to the angle, hence the sine function is used for it; the component is adjacent to the angle, hence the cosine is used for it.
Taking the signs of the components from how the vectors are drawn in the figure,

The resultant is shown in Figure 3 with a magnitude of straight up as drawn.

By symmetry the other resultant forces acting on the other two charges will have the same magnitude with a direction outward from the triangle center and below the axis as shown. In terms of absolute angles measure counter-clockwise from the axis the net forces are at , and .

## § Chapter 16, Problem 14

Given four charges at the four vertices of a square with side lengths, compute the net force on each charge. See figure 4.

Notice that there will be three forces on each charge as indicated in the figure. Two of the forces will be along the sides of the square and equal in magnitude to each other where the third will be along the diagonal. Compute the length of the diagonal using Pythagorean theorem as indicated in the drawing. Computing the magnitudes of the forces,

Working in components

To compute the resultant force,

The resultant forces on each of the four charges will have this same magnitude, all will be aligned along the diagonals and outward from the center of the square. Measured counter-clockwise from the axis they will be at , , and .

## § Chapter 16, Problem 15

Given two charges and two charges, alternating , at the four vertices of a square with side lengths, compute the net force on each charge. See figure 5.

For this problem we can use the results of Problem 14 noting that the forces aligned with the sides of the square are now attractive and hence have changed direction. Otherwise the individual forces have the same values as in Problem 14. We need to modify the sign of some vector components and recalculate the resultant.

Notice that there will be three forces on each charge as indicated in the figure. Two of the forces will be along the sides of the square and equal in magnitude to each other where the third will be along the diagonal. Compute the length of the diagonal using Pythagorean theorem as indicated in the drawing. Computing the magnitudes of the forces,

Working in components

To compute the resultant force,

The resultant forces on each of the four charges will have this same magnitude, all will be aligned along the diagonals and outward from the center of the square. Measured counter-clockwise from the axis they will be at , , and .
To compute the resultant force,

By symmetry all four forces are equal in magnitude and directed from the charge towards the center of the square.

## § Chapter 16, Problem 23

What are the magnitude and direction of the electric force on an electron in a uniform electric field of strength due east?

The charge of the electron is

The magnitude of the force is given by

The negative sign on the charge indicates that the force and the electric field are in opposite directions, hence the direction is due west.

## § Chapter 16, Problem 24

A proton is released in a uniform electric field, and it experiences a force of toward the south. What are the magnitude and direction of the electric field?

The charge of the proton is

The magnitude of the electric is given by

The positive sign on the charge indicates that the force and the electric field are in the same direction, hence the direction is south.

## § Chapter 16, Problem 25

A downward force of is is exerted on a charge. What are the magnitude and direction of the electric field at this point?

The negative sign on the charge indicate that the electric field and the force are in opposite directions, hence the field is upward. Its magnitude is

## § Chapter 16, Problem 26

What are the magnitude and direction of the electric field directly above an isolated charge?

The charge is positive, hence it acts as a source, and field lines will be directed radially outward from it. Directly above the field lines will be straight up. At the distance indicated the magnitude is

## § Chapter 16, Problem 28

What is the magnitude and direction at the midpoint between a charge and a charge separated by ?

See Figure 6. The positive charge acts as a source, and the negative charge acts as a sink. The evaluation point is from either. The fields add constructively with the resultant pointing towards the negative charge with a magnitude

## § Chapter 16, Problem 34

Calculate the electric field at one corner or a square if the other three corners are occupied by charges. See Figure 7.

Notice that there will be contributions to the total electric field from each each of three charges as indicated in the figure. Two of the contributions will be along the sides of the square and equal in magnitude to each other where the third will be along the diagonal. Compute the length of the diagonal using Pythagorean theorem as indicated in the drawing. Computing the magnitudes of the forces,

Working in components

To compute the resultant force,

counter-clockwise at from the axis which is outward from the center along the diagonal.