1. To be worked at the blackboard in lecture. R
Example: calculate the electric field at “center” of semicircular line of uniformly-distributed positive charge, oriented as shown. +Q
To be worked at the blackboard in lecture. R y x
You don’t have to follow the steps in the exact order I present here. Just let the problem tell you what to. You may do things in a different order; that’s probably OK. d ds R  dE

2. Start with our usual OSE. dq ds
Example: calculate the electric field at “center” of semicircular line of uniformly-distributed positive charge, oriented as shown. +Q
Start with our usual OSE. dq ds d R y
Pick an infinitesimal dq of charge. x
dq subtends an arc length ds,
and an angle d.
What is the charge dq?

3. Draw the dE due to the dq, and show its components. dq ds dq′ R
Example: calculate the electric field at “center” of semicircular line of uniformly-distributed positive charge, oriented as shown. +Q
Draw the dE due to the dq, and show its components. dq ds
dq′ d R y
Do you see any helpful symmetry?
dE′ dE x
Pick a dq′ horizontally across the arc from dq. The x-components of dq and dq′ will cancel. Because of this symmetry, Ex = 0
Each dEy points downward so Ey will be negative.

4.  is also one of the angles in the vector triangle.
Example: calculate the electric field at “center” of semicircular line of uniformly-distributed positive charge, oriented as shown. +Q
Recall that dq and ds are infinitesimal. dq is located at an angle  along the semicircle from the negative y-axis. dq ds dq R y  dE x
 is also one of the angles in the vector triangle. 

5. Example: calculate the electric field at “center” of semicircular line of uniformly-distributed positive charge, oriented as shown. +Q
An arc of a circle has a length equal to the circle radius times the angle subtended (in radians): dq ds dq R y  dE x 
Also,

6. Let’s summarize what we have done so far. dq ds dq Ry  dE x 
Every dq is a distance R away from the arc center:

7. Awesome Youtube derivation: http://www.youtube.com/watch?v=L1n2EUvayfw+Q dq ds dq R y  dE x 
Awesome Youtube derivation: