Question

A charged cork ball having mass $1\, g$ 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 $g=10\,m s^{-2}$)

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

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

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