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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
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