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The dipole moment of a system of charge +q distributed uniformly on an arc of radius R subtending an angle π/2 at its centre where another charge -q is :

The dipole moment of a system of charge +q distributed uniformly on an arc of radius R subtending an angle π/2 at its centre where another charge -q is :

Grade:12

1 Answers

Nehal
19 Points
4 years ago
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 styleNow, 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

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