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

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?

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.

 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. 

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,

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

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