When three charges form a triangle, a calculation of the electrostatic force requires a two-dimensional vector treatment.  This means that directions need to be expressed using either the polar method or the navigator method.

 

Read & Do

Do "Example 10.5" on page 534 in your textbook. Be sure to follow the solution carefully.

 

 

Self-Check

Answer the following self-check (SC) question to confirm your understanding of example 10.5.  When ready, click the "Check your work" bar to assess your response.

 

SC 7. 

Complete "Practice Problem" 2 on page 534 of your textbook.

 

   Self-Check Answer

SC 7.

 

Given

 

Required

to find the net force acting on sphere B, 

 

Analysis and Solution

Step 1:  Determine the direction of the forces.

 

The charge on sphere A is negative, and the charge on sphere B is positive. Therefore, the electrostatic force of  q A  on  q B  is an attractive force, pulling  q B  to the left. Similar thinking explains why the electrostatic force of  q C  on  q   B  is an attractive force, pulling  q B  towards the bottom of the page.

 

If the polar method is used, then the following free-body diagram illustrates the forces acting on  q 2  .

 

 

Note that since the spheres A and C have equal charges, and since these two spheres are the same distance from sphere B,   is the same length as   . This symmetry can be used to simplify the solution.

 

Step 2:  Calculate the magnitude of the forces acting on sphere B.

 

 

It is not necessary to do a separate calculation for the magnitude of   since the calculation will be identical to that for   . Therefore,

 

 

Step 3:  Calculate the net electrostatic force on sphere B.

 

 

At this point, it is critical to realize that this equation is describing a two-dimensional vector addition. This means that the net force will be the resultant formed by adding    and    head to tail.

 

 

Since the net force is the hypotenuse of a right-angled triangle, the Pythagorean theorem can be used to find the magnitude of   .

 

 

The fact that    and    have the same length means that the acute angle for    must be 45°. Using the polar method, the direction of    can be determined by measuring counterclockwise from the  x  -axis.

 

 

The direction of the net force is 225° using the polar method. If the navigator method were used, then south would replace the negative  y  direction and west would replace the negative  x  direction. The direction of the net force would be 45° south of west.

 

Paraphrase

The net electrostatic force acting on sphere B is 2.54 × 10 2  N [225°].

 

The previous example and practice problem show how powerful a systematic approach to problem solving can be.  Did you recognize that this method of dealing with vectors using the polar or the navigator methods stemmed directly from your work with momentum earlier in the course?  Even though the topic is no longer momentum, the same techniques work.  This isn't good luck-it is by design.  Remember, the whole idea is to use the same overall strategies throughout the course so that you can become a successful problem solver.

 

As long as you take your time and consistently follow the recommended approach, even the most complicated problems can be solved, especially if you effectively use diagrams-both vector diagrams and free-body diagrams.

 

Watch & Do

Open the multimedia learning object called Coulomb's Law in 2D and carefully watch the solution to the question.  When done, read "Example 10.6" on pages 535 to 537.  As you read, try to anticipate the next step in the solution.