Question

Four identical pendulums are made by attaching a small ball of mass $100$g on a $20$cm long thread and suspended from the same point. Now each ball is given charge Q so that balls move away from each other with each thread making an angle of ${45}^o$ from the vertical. The value of Q is close to ($\frac{1}{4\pi {ϵ}_0}=9\times {10}^9$ in SI units).

• A. $1\mu$C
• B. $1.5\mu$C
• C. $2\mu$C
• D. $2.5\mu$C

Answer 1

Answer: (B)

$1.5\mu$C

From the diagram,

$OA=L/\sqrt{2}$

$⟹AC=BC=L$

$AD=L\sqrt{2}$

Net electrical force on charge A is:

$F=F_{AD}+F_{AB}\cos {45}^o+F_{AC}\cos {45}^o$

$F=\frac{Q^2}{4\pi {\epsilon }_o\times 2L^2}+\frac{Q^2}{4\pi {\epsilon }_o{L}^2\sqrt{2}}+\frac{Q^2}{4\pi {\epsilon }_o{L}^2\sqrt{2}}$

$F=\frac{\sqrt{2}{Q}^2}{16\pi {\epsilon }_o{L}^2}(2\sqrt{2}+1)$

Perpendicular to the string, net force is zero.

$F\sin {45}^o=mg\sin {45}^o$

$\frac{\sqrt{2}{Q}^2}{16\pi {\epsilon }_o{L}^2}(2\sqrt{2}+1)=mg$

$Q=1.61\times {10}^{-6}C$