  Question

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

A
22qRπ  B
2qRπ  C
qRπ  D
2qRπ  Solution

The correct option is B $$\displaystyle \frac{2\sqrt{2}qR}{\pi}$$The definition of dipole moment is $$\vec{p}=\int \vec{r}dq$$If $$\lambda$$ is the line charge density, charge on element $$dl$$ is $$dq=\lambda Rd\theta$$In polar coordinate,  $$\vec{r}=R\cos\theta \hat i+R\sin\theta \hat j$$thus, $$\displaystyle \vec{p}=\int (R\cos\theta \hat i+R\sin\theta \hat j)\lambda Rd\theta$$$$\displaystyle p_x=\lambda R^2\int_0^{\pi/2}\cos\theta d\theta=-\lambda R^2$$$$\displaystyle p_y=\lambda R^2\int_0^{\pi/2}\sin\theta d\theta=\lambda R^2$$thus, $$\displaystyle p=\sqrt{p_x^2+p_y^2}=\sqrt{2}\lambda R^2=\sqrt{2}R^2 \frac{2q}{\pi R}$$  $$\displaystyle (q=\int_0^{\pi/2}\lambda Rd\theta=\frac{\lambda R\pi}{2})$$$$\displaystyle \therefore p=\frac{2\sqrt{2}qR}{\pi}$$ PhysicsNCERTStandard XII

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