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Determining electric field at origin from semicircle charge

  1. Oct 5, 2016 #1
    1. The problem statement, all variables and given/known data
    A line of charge of uniform density ρl occupies a semicircle of radius b as shown in Fig. P4.10. Use the material presented in Example 4-4 to determine the electric field at the origin.


    2. Relevant equations
    dE = kdQ/r^2

    3. The attempt at a solution
    I'm trying to understand the solution given in the solutions manual, here it is:

    http://www.zmuda.ece.ufl.edu/Fields_1_Spring 2013/4-1.10.pdf

    1. Why is the direction -r hat, rather than just r hat?
    2. Why is the enclosed charge Q stated as (pl)(b)? Shouldn't it be pl multiplied by the arc length of the half circle? I understand that I'm wrong and the solution is right, but I do not understand why the charge density would be multiplied by the radius.
     
  2. jcsd
  3. Oct 5, 2016 #2

    olivermsun

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    Science Advisor

    1. The electric field felt at the origin due to the charge ##dE## at position ##\hat{\mathbf{r}}## points from ##\hat{\mathbf{r}}## toward the origin. (Recall that a positive ##dE## would exert a repulsive force on a positive test charge at the origin.)

    2. What is the arc length at radius ##b## that subtends the angle ##d\phi##?
     
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