Poll result: direction of electric field due to point charges

This problem originated with an exercise in Serway, but I've made an important change.  I'll explain that change at the end of this post.  First, answer the question:

The blue charge at the top left produces an electric field that points to the left (toward the negative charge producing the field).  The blue charge at the bottom right produces an electric field that points down the page. 

So the vector sum of the electric fields produced by the BLUE charges is 45 degrees down and to the left. 

The red charge produces an electric field that points up and to the right (away from the positive charge). 

The total electric field now is the vector sum of the fields due to the blue charge, and that due to the red charge.  That total electric field must point in either direction (a) or direction (d) as indicated in the diagram.  Sure enough, the vast majority of poll respondents were split between these two directions.  Which is correct?

The answer hinges on which electric field is bigger:  the one due to the red charge, or the one due to the blue charges?

Each blue charge has the same size charge, and each is the same distance from the position where we're measuring the field.  So each blue charge produces a field with the same magnitude; by vector addition, the total field due to the blue charges will be (root 2) times the field produced by one of the blue charges.

The red charge has the same size charge as the blue charges, but is a larger distance from the relevant position.  So the red charge must produce a field that is SMALLER than that produced by either blue charge.

The field due to the blue charges is larger than the field due to the red charges, and pointing in the opposite direction.  So the vector sum of all these electric fields is pointing in the direction of the larger field, the one from the blue charges, the one in the direction of choice (d). 

Now for the teaching point.  Why did I change Serway's diagram?  The original problem had the red charge as +2Q.  Well, in that case you'd need to think carefully about the red electric field.  The charge producing the field is farther away than the blue charges, but the charge itself is bigger.  To get the answer right, you need to plug carefully into E=kQ/d².  This problem is designed just to check my students' conceptual understanding of electric fields due to point charges -- I want to start simple.  Once everyone can merely find the correct direction of the electric field, once everyone can understand conceptually that a charge farther away from a field point produces a smaller electric field at that point, then we can start doing more complicated gemoetry and algebra.

GCJ

Multiple Choice poll: electric field due to point charges

The poll on the left is based on a picture from Serway.  I've changed it a bit, and not just so the Serway Lawyers don't come after me -- I've made the question more conceptual than mathematical.  Go ahead and vote.  I'll post results next weekend, along with a discussion of the change I made and why I made it.

FYI, my AP class has just begun studying electric fields due to point charges.  This is their first homework problem, with the request for a justification of the answer, of course.

GCJ

Electrostatic point charges ranking task poll -- Here's the answer!

Electric field is a vector -- that means that a sum of many electric fields requires vector addition, with arrows.  Just adding up the values of kQ/d^2 does not work.  (I don't care how many times you say this, someone in your class will forget on a test or quiz.)

The electric field created by a point charge points (hah!) away from a positive charge, and toward a negative charge.

Electric field vectors, like all vectors, are best added by placing them tip-to-tail.

Consider diagram 1 in the ranking task.  The top charge creates a field to the right (away from the charge); the bottom charge creates a field up and to the right.  The vector sum of these electric fields is in red below:

DIAGRAM 1



Now consider diagram 2.  The top charge still creates a field to the right, but the bottom charge now creates a field down and to the left (toward the negative charge).  The vector sum of these fields is in red below:

DIAGRAM 2

From the picture, it is apparent that the magnitude of the electric field in diagram 1 is bigger than the magnitude of the electric field in diagram 2 -- the arrow for Etotal is bigger.

Similar reasoning for diagrams 3 and 4 gives the same sized total electric field vectors, but pointing the opposite directions.  Since the *magnitude* of an electric field means the amount of electric field, regardless of direction, we can say that diagram 3's field has identical magnitude to diagram 1's field; and that diagram 2's field is equal in magnitude to diagram 4's field.

Thus the answer:  (1=3), (2=4).

You wanna rank the voltages at point P?  Go ahead, post a comment.

Poll Answer: Electric Field Created by a Charge

The poll question stem, which was posted throughout the week, is shown to the right.  Here's my answer.

Let's start with direction.  The electric field produced by a positive charge points away from the charge; the electric field produced by a negative charge points toward the charge.  Since a negative charge produces this electric field, the direction is "toward the charge."

Now for the magnitude of the electric field.  First of all, our students must be taught to use the variables given in the problem.  I know that the textbook says that the electric field due to a charge is E = kq/r^2.  This problem defined the distance from the charge as "L".  Thus, we must use "L" and not "r" in the answer.

Secondly, our students must understand what the word "magnitude" means.  All vector quantities, including electric field, include an amount of something (the magnitude) and a direction.  For example, a velocity might be 30 m/s toward the south.  In this case, "30 m/s" is the velocity's magnitude, and "south" is the direction. 

Now, when a vector is constrained to one direction (or when we're dealing with components of a 2-d or 3-d vector), it is mathematically convenient to define a negative and a positive direction.  If I'm working a kinematics problem with this car, I might call north the positive direction, in which case the velocity vector can be plugged in simply as "-30 m/s".  But, importantly, the magnitude of this vector is still "30 m/s."  The magnitude of a vector can not be negative!

Electric field is a vector.  When using equations regarding electric fields, never plug in the sign of the charge!  use the equation to find the magnitude of an electric field or force.  Then, use memorized facts to determine the direction of an electric field or force. 

In this case, then, the negativeness of the charge producing the electric field does not affect the field's magnitude.  The field has magnitude kq/L^2.  The direction of the electric field is toward the charge.

I like the poll... New question coming soon!

GCJ

Leading questions about the electric field due to point charges

First of all, this blog is up to six -- count 'em, six -- followers.  Woo-hoo!  Only a few million more and I'm as popular as Bill Simmons.  In all seriousness, I'm glad to have everyone aboard.

The diagram at the right is modified a bit from a Serway electrostatics problem.  The goal in the problems I assign, as you may have guessed, is to find the electric field and the electric potential at point P due to the charges.  Today, I just want to talk about finding the electric field.

I will ask this question differently depending on how long we have been studying electrostatics. 

If it is our first foray into fields due to point charges, I will start with:

i.  Is there a charge at point P?

ii.  Is there a force at point P?

iii.  Is there an electric field at point P?

Don't think that's so silly.  I guarantee you, if you don't start here, a large number of students will assume that point P is "positive," or consists of an electron, or will be confused by the sign of point P, or something like that.  Be sure that the class understands the difference between a point charge and a point in space.

If we've had an introduction to fields produced by charges, I might ask the following series of leading questions:

(a) What is the magnitude and direction of the electric field at point P due to the top-left charge?

(b) What is the magnitude and direction of the electric field at point P due to the bottom-right charge?
(c) Calculate the magnitude and direction of the electric field at point P due to BOTH the top-left and bottom-right charges.
(d) What is the magnitude and direction of the electric field at point P due to the bottom-left charge?
(e) Calculate the net electric field at point P. Include both magnitude and direction.

And only after we have gone through example after example in class and on homework would I ask the (nearly) full monty:

1.  Calculate the magnitude and direction of the electric field at point P.

The final, culminating question, which I dread to ask, but I will ask after we are absolutely clear on how to calculate the electric field at point P:

I.  Consider an electron placed at point P.  Calcualte the magnitude and direction of the force experienced by this electron due to the surrounding charges.

Electric fields and potentials of point charges

The electric FIELD produced by a point charge is E = kQ/d². The electric POTENTIAL produced by a point charge is V = kQ/d.

Electric field is a vector. This means that the equation E = kQ/d² gives just the magnitude of the electric field, and therefore the sign if the charge producing the field should NOT be plugged into the equation. The direction of the field is away from a positive charge, and toward a negative charge.

Electric potential is a scalar. This means that the sign of the charge producing the potential SHOULD be plugged into the equation V = kQ/d. Positive charges produce positive potentials; negative charges produce negative potentials.

When more than one charge produces an electric field, the net field is found by vector addition. When more than one charge produces an electric potential, the net potential is found by algebraic addition and subtraction.

I can’t begin to tell you how frustrating it is to go over the above paragraphs 30 times in 15 different ways with 80 different examples… and then to see someone tell me that the electric field due to a couple of charges is “(2-Q)/d.” Aargh!!!

Such is the nature of electricity, though. I have to remind myself time and again how abstract the idea of a “field” is to begin with, let alone potential, the concept of “charge,” the creation of a field or potential… double Aargh!

The only success I’ve found at the physics B level with these topics is through repetition. There is no choice but to ask students quiz questions 30 times in 15 different ways with 80 different examples. If that doesn’t work, give it a rest for a week… then ask a 31st time.

Below are four questions from a fundamentals quiz which get to the heart of electric fields and potentials due to point charges. Can you answer them correctly? (Post a comment with your answer! If you’re wrong, that’s okay… someone will correct you.)

1. What is the electric potential produced by a -3Q charge a distance of 2a away from the charge?

2. What is the magnitude of the electric field produced by a -3Q charge a distance of 2a away from the charge?

3. In the diagram to the right, what is the magnitude of the electric field produced by the two charges at point P?

4. In the diagram to the right, what is the vertical component of the electric field produced by the –Q charge at point P?

GCJ

(Photo at the top from alexhulbert.com . )