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Two identical charged spheres suspended from a common point by two massless strings of lengths \l, are initially at a distance d (d << \l) apart because of their mutual repulsion. The charges begin to leak from the both the spheres at a constant rate. As a result, the spheres approach each other with a velocity \nu. Then \nu varies as a function of the distance x between the spheres as:

  • Option 1)

    \nu\propto x^{\frac{1}{2}}

  • Option 2)

    \nu\propto x

  • Option 3)

    \nu\propto x^{-\frac{1}{2}}

  • Option 4)

    \nu\propto x^{-1}

 
Answers (1)
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As discussed in

Coulombic force -

Fpropto Q_{1}Q_{2}=Fpropto frac{Q_{1}Q_{2}}{r^{2}}=F=frac{KQ_{1}Q_{2}}{r^{2}}

- wherein

K - proportionality Constant 

Q1 and Q2 are two Point charge

 

 T \cos \Theta=mg\:\:\:\:-(i)

T \sin \Theta=qt\:\:\:\:\:-(ii)

\tan \Theta=\frac{qt}{mg}=\frac{Kq^{2}}{x^{2}mg}

\frac{x}{2l}=\frac{Kq^{2}}{x^{2}mg}\:\:\:\:\Rightarrow q^{2}=\frac{x^{3}mg}{2lK}\:\:\:or \:\:\:q\propto x^{\frac{3}{2}}

\frac{dq}{dt}\propto \frac{3}{2}\sqrt x\:v

Where \frac{dq}{dt}=constant

\therefore V \propto\frac{1}{\sqrt x}=V\propto x ^{\frac{-1}{2}}

 


Option 1)

\nu\propto x^{\frac{1}{2}}

This option is incorrect.

Option 2)

\nu\propto x

This option is incorrect.

Option 3)

\nu\propto x^{-\frac{1}{2}}

This option is correct.

Option 4)

\nu\propto x^{-1}

This option is incorrect.

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