Charge q is distributed uniformly on an arc of radius R subtending right angle at its centre.Another charge -q is placed at the centre of arc.Find dipole moment of system??

consider a small element on arc of length dl substending d@ at the center ...

let angle made with base is @ ...this is placed in XY plane with base in x axis ...

charge of dl element = dq

dq = pRd@             (p is charge dencity)

let us assume that central charg  divided into n parts of dq magnitude then every dq element of arc 

& central charge will constitute a dipole ...

dP (dipole moment) = q(2a) = dq R

dP = dqR² [ cos@(i) + sin@ (j) ] lim from 0 to pi/2

dP = pR²[ cos@d@(i) + sin@d@(j) ]      lim 0 to pi/2

P(net dipole moment) = pR² [ i + j ]

now , p = q/(piR/2) = total charge/length of arc so

Pnet = 2qR/pi . [ i + j]   with direction

Approved