Physics 230 February 11 2002 Problem: A rod of length 2a lies along the x-axis with its center at the origin. It has a uniform charge density l (= Q/2a). Find the electric field at a point P on the x-axis with x-coordinate xP.

Start with a diagram. Show the rod, the coordinate axes, and the point P. Make your diagram BIG.
 
 

Step I: Model the system (the rod) as a collection of differential elements. Draw a typical element on your diagram. Each element should correspond to a differential change in one coordinate.

See diagram above

Step II. Identify a typical element and describe it using your coordinates.

The element extends from x to x+dx and has charge dq = l dx

Step III. Express the contribution of your element to the desired sum. (i.e. find the electric field dE produced by this element. Give both its magnitude and its direction. With vectors, it is often easier to calculate each component separately.

As we can see from the diagram, the electric field is in the x-direction, and from Coulomb's law
 
 

 Step IV. Find the limits of the integral.

The limits are x = -a to x = a
 
 

Step V: Integrate!


 
 

Analysis: Show that your result reduces to kQ/xP2when P is a long way from the origin.
  For xP>>a, the denominator reduces to xP2, and since Q = 2al , we get the expected form of Coulomb's law.