Challenge Problem: Line of Charge
Click here to see the solution

     In the previous presentation we determined the E-Field caused by a "line of charge" with uniform charge density for a point at a distance from the end of the line of charge.  You may recall that the equation for the E-field is E = kQ/a(a+L). "a" is the distance from a point in space along the axis of the line to the end of the line of charge.
     In this challenge problem you are to imagine that the line of charge is now vertically oriented and has a length of L. Extended away from the midpoint of the line, to the right, is point P (a distance "r" away from the line of charge). Your challenge is to determine the equation for the E-field in this case. Once you determine the answer consider how the solution might simplify for two special cases: 1) if the length of the line was very long (approaching infinitely long) and 2) if the length of the line was very short in comparison to r. Use the same types of procedures you saw in the previous presentation. The difference in difficulty will lie in solving the integration, which is a bit more complex than the ones we did in the previous presentation. 
     It's important that you know the answer to special case #1 mentioned above, as we will compare it to an alternative solution you will see in the next presentation.  So if you run into difficulty solving this, don't worry...but record the answer in your notebook at a minimum.

This is not an easy problem...so if you need some clues, or need to see the solution, click here.