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 $\boldsymbol{\theta}=37^{\circ}$, when value of electric field is \(\mathbf{E}=(3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}) \times 10^5…

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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 $\boldsymbol{\theta}=37^{\circ}$, when value of electric field is $\mathbf{E}=(3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}) \times 10^5 \mathrm{NC}^{-1}$. (Assume, $T$ as tension in the string.) Which of the following options are correct? (Given, $\sin 37^{\circ}=0.60$ and $g=10 \mathrm{~ms}^{-2}$ )

Aicharya · Accepted Answer

Mass of cork ball, $m=1 \mathrm{~g}=10^{-3} \mathrm{~kg}$ $\begin{aligned} & \mathbf{E}=(3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}) \times 10^5 \mathrm{NC}^{-1} \ & \theta=37^{\circ} \end{aligned}$ Force on cork ball due to electric field $\mathbf{E}$ $\mathbf{F}=q \mathbf{E}$ According to given diagram, resolving all forces in two given direction, $\begin{array}{ll} & T \sin \theta=q E_x \quad \ldots (i) \ & T \cos \theta+q E_y=m g \ & T \cos \theta=m g-q E_y \quad \ldots (ii) \end{array}$ From Eqs. (i) and (ii), we have $\frac{T \sin \theta}{T \cos \theta}=\frac{q E_x}{m g-q E_y}$ $\begin{aligned} & \tan \theta=\frac{q E_x}{m g-q E_y} \ & \Rightarrow \quad \tan 37^{\circ}=\frac{q \times 3 \times 10^5}{10^{-3} \times 10-q \times 5 \times 10^5} \ & \frac{3}{4}=\frac{3 q \times 10^5}{10^{-2}-5 q \times 10^5} \ & \Rightarrow \quad 3 \times 10^{-2}-15 q \times 10^5=4 \times 3 q \times 10^5 \ & 0.03-15 \times 10^5 q=12 \times 10^5 q \Rightarrow q=11 \times 10^{-8} \mathrm{C} \end{aligned}$ From Eq. (i), we get $\begin{aligned} & T \sin 37^{\circ}=1.1 \times 10^{-8} \times 3 \times 10^5 \ & T \times 0.6=3.3 \times 10^{-3} \ & T=\frac{3.3}{0.6} \times 10^{-3}=5.5 \times 10^{-3} \mathrm{~N} \end{aligned}$

A charged cork ball having mass \(1 g\) and charge \(q\) is suspended on a light string in a uniform…

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