the dipole moment of a system of charge q distributed uniformly on an arc of radius R subtending an angle 90 degree at the center where another charge –q is placed as

a) 2*1.41 qR / pi

b)1.41 qR / pi

c) qR/ pi

d) 2qR/ pi

Asked by thakurpratyush5 | 24th May, 2015, 12:06: PM

Expert Answer:

The arc is subtending an angle of 90 degrees with the centre. Hence, it forms a quarter semicircle.
 
So, the charge density of the arc is begin mathsize 14px style straight rho equals fraction numerator straight q over denominator arc space length end fraction equals fraction numerator straight q over denominator begin display style πR over 2 end style end fraction equals fraction numerator 2 straight q over denominator πR end fraction end style
Now, dipole moment is given as a product of charge and distance between the two charges.
 
Hence, if we consider a length element dx of the arc, then we have
 
begin mathsize 14px style straight p equals integral subscript 0 superscript straight R ρR d straight x equals integral subscript 0 superscript straight R fraction numerator 2 straight q over denominator πR end fraction straight R d straight x equals fraction numerator 2 straight q over denominator straight pi end fraction integral subscript 0 superscript straight R dx equals fraction numerator 2 straight q over denominator straight pi end fraction open square brackets straight R close square brackets subscript 0 superscript straight R equals fraction numerator 2 qR over denominator straight pi end fraction end style

Answered by Romal Bhansali | 24th May, 2015, 07:24: PM