Students will expand their ideas about forces from multiple charges by considering the arrangement of charges at right angles.

Because forces are vectors (values with magnitude and direction), they must be added geometrically.

15 minutes

Today's warmup is relatively simple as it features the now-familiar set of three collinear charges. As a twist, I ask students to identify the net force on one of the outside charges. The goal here is to continue to build facility with Coulomb's Law and to remind students that all charges exert forces on all other charges. I want students to be facile with this kind of thinking so that, later in this lesson, we can determine how to proceed with net force calculations when forces are at right angles.

Students may work solo or with partners. I circulate, assisting those most in need and affirming the work of those who are on track. After five minutes or so, I reveal the solution at the Smartboard and spend a few minutes highlighting important recognitions. For example, the distance from Q1 to Q3 is not given explicitly but can be easily determined from the distances that are provided.

30 minutes

My content goal here is to have students become adept at handling forces at right angles. To do so requires an extended set of ideas that takes some time. Though *no single step* in the process is too challenging, there are a significant *number* of steps and students can lose the trail easily. My lecture proceeds slowly, with many check-ins along the way. Students take notes and participate in a dialog with me as we proceed.

To start, I use a set of forces that are at right angles and ask students to predict the *general direction* of the net force (see introduction to forces at right angles). Most will intuitively feel that the net force will be at an angle (say, to the northeast for the combination of a north force with a east force). I share with them that our technique will allow us to predict the exact direction of the net force.

The new ideas presented during this segment of class are numerous:

1) Force are vectors which have magnitude and direction.

2) Vectors are added "tip-to-tail" and, in these cases, form two parts of a right triangle.

3) The net force is the hypotenuse, determined by the Pythagorean Theorem.

4) The angle is determined by using the inverse tangent function.

I show the process of vector addition, then a summary of right angle mathematics.

I begin by using a problem based on forces at right angles, without the complication of electric charges. I demonstrate a solution then reconsider the first step of the problem and ask: What would we need to do if we began with *three charges* at right angles? Students respond that we must first calculate the forces based on charges and distances and determine directions based on the signs of the charges. One hint I provide to students is to determine the directions of the forces first, then create a right triangle that has the correct orientation. After that, they merely have to invoke Coulomb's Law to get the precise values of the forces before proceeding.

25 minutes

I provide students with three right-angle charge problems, each of which features three charges at right angles and for which the challenge is to find the net force acting on Q2, the charge at the right angle vertex. Before allowing students to work independently, I spend just a bit more time working through the first of these three problems to create a complete picture of this kind of problem. Half my students calculate one of the forces on Q2 while the other half calculates the second force. Within a minute or two, we are dealing with the right triangle nature of the forces involved. I demonstrate the overall solution, then encourage students to try the remaining two problems.

Students are welcome to collaborate as I circulate from student to student to observe the level of understanding. This time should be very formative and, while students may be anxious about the content, they should not be anxious about being evaluated.

10 minutes

I use the final few minutes to change directions today as students often find this lesson to be difficult. The burden shifts to me during these minutes and students can "let down" a bit. I choose to review a problem from the previous lesson as very few student teams were able to make progress with it.

I quickly work the three-charge problem at the board, but delay inserting any numbers until the very end. By doing so, I hope to demonstrate that some terms will cancel and leave one with a much simpler set of numbers to deal with. Given the amount of time left in class, the solution is incomplete; I simply want to show students the value of keeping the problem statement algebraic for as long as possible.