Question

A charged cork ball having mass $1g$ and charge ' $q$ ' is suspended on a light string in a uniform electric field as shown in figure. The ball is in equilibrium at $\theta =37^{\circ }$, when value of electric field is $\vec{E}=(3\hat{i}+5\hat{j})\times 10^{5}N\cdot C^{-1}$. Assume $T$ as tension in the string. Which of the following options are correct ? (given : $\sin 37^{\circ }=0.60$ and $g=10m{s}^{-2}$)

• A. $q=11\times 10^{-8}C$
• B. $T=5.55\times 10^{-3}N$
• C. $q=12\times 10^{-9}C$
• D. $T=4.55\times 10^{-3}N$

Answer 1

Answer: (B)

Mass of cork ball, $m=1g=10^{-3}kg$
$E=(3\hat{i}+5\hat{j})\times 10^{5}N{C}^{-1}$
$\theta =37^{\circ }$

Force on cork ball due to electric field $E$
$F=qE$
According to given diagram,
resolving all forces in two given direction,
$T\sin \theta =q{E}_x...(i)$
$T\cos \theta +q{E}_y=mg$
$\Rightarrow T\cos \theta =mg-q{E}_y...$ (ii)
From Eqs. (i) and (ii), we have
$\frac{T\sin \theta }{T\cos \theta }=\frac{q{E}_x}{mg-q{E}_y}$

$\tan \theta =\frac{q{E}_x}{mg-q{E}_y}$
$\frac{3}{4}=\frac{3q\times 10^5}{10^{-2}-5q\times 10^5}$
$\Rightarrow 3\times 10^{-2}-15q\times 10^5$
$=4\times 3q\times 10^5$
$0.03-15\times 10^{5}q=12\times 10^{5}q$
$\Rightarrow q=1.1\times 10^{-8}C$
From Eq. (i), we get
$T\sin 37^{\circ }=1.1\times 10^{-8}\times 3\times 10^5$
$T\times 0.6=3.3\times 10^{-3}$
$T=\frac{3.3}{0.6}\times 10^{-3}$
$=5.5\times 10^{-3}N$