Consider a circular arc of radius `R` which subtends an angle `phi` at its centre .Let us calculate the electric field strength at `C`

Explanation

Consider a polar segment on arc of angular width `d theta` at an angle `theta` from the angular bisector XY as shown. The length of elemental segment is `Rd theta`. The charge on this segment dq is
`dq=Q/(phi) d theta`
Due to this dq, electric field at centre of arc C is given as
`dE=(dq)/(4 pi epsilon_(0) R^(2))`
The electric field component dE to this segment dE `sin theta` which is perpendicular to the angle bisector gets cancelled out on integration.
The net electric field at centre will be along angle bisector which can be calculate by integrating `dEcos theta` within limits from `-phi//2` to `phi//2`
Hence net electric field strength at centre C is
`E_(c)=int d E cos theta`
`=int_(-phi//2)^(phi//2)Q/(4 pi epsilon_(0) phiR^(2)) cos theta d theta`
`=Q/(4 pi epsilon_(0) R^(2) phi)int_(-phi//2)^(phi//2) cos theta d theta`
`=Q/(4 pi epsilon_(0)R^(2) phi)[sin theta]_(-phi//2)^(phi//2)`
`Q/(4 pi epsilon_(0) R^(2)phi)[ sin phi //2 + sin phi//2]`
`E_(c)=(2 Q sin (phi//2))/(4pi epsilon_(0)R^(2) phi)`
For a semi circular ring `phi=pi`. So at centre
`E_(c)=(2Qsin(phi///2))/(4 epsilon_(0)R^(2) phi)=(2Qsin (pi//2))/(4 pi epsilon_(0)R^(2)pi)=(2Q)/(4 pi^(2) epsilon_(0)R^(2))`