Question

One end of $10cm$ long silk thread is fixed to a large vertical surface of a charged nonconducting plate and the other end is fastened to a small ball having a mass of $10g$ and a charge of $4.0\times {10}^{-6}C$. In equilibrium, the thread makes an angle of ${60}^o$ with the vertical. Find the surface charge density on the plate.

Answer 1

Ball mass $=10g$

Charge $=4\times {10}^{-6}C$

Thread length $=10cm$

Now from the figure, $T\cos \theta =mg$

$T\sin \theta =$ electric force

Electric force $=\frac{\sigma q}{2{\in }_0}$ ($\sigma$ surface charge density)

$T\sin \theta =\frac{\sigma q}{2{\in }_0},T\cos \theta =mg$

$\tan \theta =\frac{\sigma q}{2mg{\in }_0}$

$\sigma =\frac{2mg{\in }_0\tan \theta }{q}=\frac{2\times 8.85\times {10}^{-12}\times 10\times {10}^{-3}\times 9.8\times 1.732}{4\times {10}^{-6}}=7.5\times {10}^{-7}C/m^2$