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17 January 2011

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

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