Question 701: Electrostats & Capacitance

Question

A point charge $q=1 \mu \mathrm{C}$ is located at a distance 2 cm from one end of a thin insulating wire of length 10 cm having a charge $Q=24 \mu \mathrm{C}$, distributed uniformly along its length, as shown in figure. Force between $q$ and wire is $\_\_\_\_$ N.
(Use : $\frac{1}{4 \pi \epsilon_0}=9 	imes 10^9 \mathrm{~N} \cdot \mathrm{~m}^2 / \mathrm{C}^2$ )

Accepted Answer

Because the charge on the wire is distributed, we consider a small element dx on the wire at a distance x from the point charge q.
Linear charge density of the line charge is $\lambda=\frac{\mathrm{Q}}{\mathrm{L}}$
Charge on element $\mathrm{dQ}=\lambda \mathrm{dx}$
Small force acting between the charges is,
$$ \mathrm{dF}=\frac{1}{4 \pi \epsilon_{\mathrm{o}}} \frac{\mathrm{q} \cdot \mathrm{dQ}}{\mathrm{x}^2}=\frac{\mathrm{kq} \mathrm{q} \lambda \mathrm{dx}}{\mathrm{x}^2} $$
To find the total force F , we integrate from the nearest end $(\mathrm{x}=2 \mathrm{~cm}=0.02 \mathrm{~m})$ to the farthest end ( $\mathrm{x}=2 \mathrm{~cm}+10 \mathrm{~m}=12 \mathrm{~cm}=0.12 \mathrm{~m}$ ):
$$ \mathrm{F}=\int\limits_{0.02}^{0.12} \frac{\mathrm{kq} \lambda}{\mathrm{x}^2} \mathrm{dx}=\mathrm{kq} \lambda\left[-\frac{1}{\mathrm{x}}ight]_{0.02}^{0.12} $$
$$\Rightarrow $$ $\mathrm{F}=\mathrm{kq} \lambda\left(\frac{1}{0.02}-\frac{1}{0.12}ight)=\mathrm{kq} \lambda\left(\frac{6-1}{0.12}ight)=\frac{5 \mathrm{kq} \lambda}{0.12}$
$$\Rightarrow $$ $\mathrm{F}=\frac{5 \mathrm{kq}}{0.12} 	imes \frac{\mathrm{Q}}{\mathrm{L}}$
Substituting the given values,
$$\mathrm{k}=9 	imes 10^9 ; \mathrm{q}=10^{-6} ; \mathrm{Q}=  24 	imes 10^{-6} ; \mathrm{L}=10 \mathrm{~cm}=0.10 \mathrm{~m} $$
$$  \mathrm{~F}=\frac{5 	imes 9 	imes 10^9 	imes 1 	imes 10^{-6} 	imes 24 	imes 10^{-6}}{0.10 	imes 0.12} \mathrm{~N} $$
$\begin{aligned} \Rightarrow \mathrm{F} & =\frac{216}{2.4} \ \Rightarrow \mathrm{~F} & =90 \mathrm{~N}\end{aligned}$
Therefore, the electrostatic force exerted by the finite wire on the point charge is 90 N .
Hence, the correct answer is 90 .

Electrostats & Capacitance

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