Help me out{Electric field intensity}

well i gave another question!!!

A semicircular ring of mass m is kept in x-y plane with centre at origin. It
is charged in such a way such that charge per unit length varies as λ=λocosθ (θ is angle measured w.r.t. positive x-axis!).
Find torque about the centre of ring when electric field E=E_oxi+E_oj is just switched on.

ADDITION:: Find torque about centre of half ring!!

Let me try this...

the applied field= E_oxi + E_oj

so constant field applies to y direction, and by symmetry will exert NO NET FORCE on the half ring of charge...

so the applied field is equivalent to E_o . x . i only!!

referring to Sanchit's diagram,

considering a small arc subtending angle d@ as in figure, and consider the charge distribution is uniform in this region...

length=Rd@

linear charge density=λ=λ_ocos@

so charge on the small element=Rλ_ocos@d@

x coordinate=R cos@

so, field=E_oRcos@

so force on small element = E_oλ_oR²cos²@d@

so net force=\int_{0}^{\pi }{E_o \lambda _oR^2cos^2\theta d\theta }=\frac{E_o \lambda _oR^2}{2} \int_{0}^{\pi }{(cos2\theta+1) d\theta} = \frac{E_o \lambda _oR^2}{2}.\left[0+\pi -0-0 \right]=\frac{\pi E_o \lambda _oR^2}{2}

thus, acceleration=\frac{\pi E_o \lambda _oR^2}{2m}