Two small spheres of each charge $q$, mass $m$ and material density $d$ are suspended from a fixed point with the help of inextensible light thread. When the spheres are in air, the angle between the threads is $90^{\circ}$. When the spheres are suspended in a liquid of density $\frac{2}{3} d$,…

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Two small spheres of each charge $q$, mass $m$ and material density $d$ are suspended from a fixed point with the help of inextensible light thread. When the spheres are in air, the angle between the threads is $90^{\circ}$. When the spheres are suspended in a liquid of density $\frac{2}{3} d$, the angle between the threads is $60^{\circ}$. The value of dielectric constant of the liquid is

Aicharya · Accepted Answer

Key Idea At any instant, $ T \cos 	heta=m g 	ext { and } T \sin 	heta=F_e \Rightarrow 	an 	heta m g=F_e $ $ \begin{aligned} F_e & =\frac{k q^2}{r^2} \ r & =2 l \cos 	heta \ 	an 	heta & =\frac{F_e}{m g} \end{aligned} $ [from Fig.] So, for $	heta=45^{\circ}$ and $\phi=45^{\circ}$ $ \begin{aligned} r & =\frac{2 l}{v^2}=\sqrt{2 l} \Rightarrow F_e=\frac{K q^2}{2 l} \ 	an 45^{\circ} & =\frac{k q^2}{2 l m g} \quad\left(\because 	an 45^{\circ}=1ight) \end{aligned} $ For, $	heta=30^{\circ}$ and $\phi=60^{\circ}$ $ r=2 l 	imes \frac{1}{2}=l $ So , $F_e^{\prime}=\frac{k q^2}{l^2} \Rightarrow 	an 30^{\circ}=\frac{k^{\prime} q^2}{l^2-m^{\prime} g}$ As, the sphere is suspended in a liquid of density $\frac{2}{3} d$, then the observed weight of the body, $m^{\prime}=V\left(d-\frac{2 d}{3}ight)=\frac{m}{3} \quad[\because m=V \cdot d]$ So, from Eq. (i) and (ii), we get $ \begin{aligned} 3 \sqrt{3} k^{\prime} & =\frac{k}{2} \Rightarrow k^{\prime}=\frac{1}{4 \pi \varepsilon_0 \varepsilon_r}, K=\frac{1}{4 \pi \varepsilon_0} \ \Rightarrow \quad \varepsilon_r & =6 \sqrt{3} \end{aligned} $

Two small spheres of each charge $q$, mass $m$ and material density $d$ are suspended from a fixed point…

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