Question

One end of a string is fixed with a wall and the other end has a point charge $-10\text{nC}$. It is kept along the axis of a finite line charge as shown in figure. If string is plucked, then the speed of the wave generated will be:

• A. $268\text{m/s}$
• B. $278\text{m/s}$
• C. $288\text{m/s}$
• D. $298\text{m/s}$

Answer 1

The correct option is A $268\text{m/s}$

Electric field due to the line charge at the end of the string having charge $-10\text{nC}$,
$E=k\lambda [\frac{1}{r}-\frac{1}{r+L}]$

Given:
$r=1\text{m}$
$L=4\text{m}$
$\lambda =10\text{C/m}$
$q=-10\text{nC}$
$\mu =0.01\text{kg/m}$

$\Rightarrow E=9\times {10}^9\times 10\times [\frac{1}{1}-\frac{1}{1+4}]$
$\Rightarrow E=7.2\times {10}^{10}\text{N/C}$

Now, from equilibrium of forces for the point charge,
Tension, $T=qE$
$\Rightarrow T=10\times {10}^{-9}\times 7.2\times {10}^{10}$
$\Rightarrow T=720\text{N}$

Speed of wave in the string, $v=\sqrt{\frac{T}{\mu }}$
$\Rightarrow v=\sqrt{\frac{720}{0.01}}$
$\Rightarrow v=268\text{m/s}$