Determining electric field at origin from semicircle charge

[JesseJC]

Homework Statement

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.

Homework Equations

dE = kdQ/r^2

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.

 

[olivermsun]

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 $d\phi$?