Two identically charged pith balls are suspended from the some point by two massless identical threads density of each ball is $\rho $. If system is immersed in a medium of density $\sigma $, balls remain undeflected, then the dielectric constant of medium is

  • A

    $\frac{\rho }{{\rho  - \sigma }}$

  • B

    $\frac{ \rho  - \sigma }{{\rho }}$

  • C

    $\frac{\sigma }{{\rho  - \sigma }}$

  • D

    $\frac{ \rho  - \sigma }{{\sigma }}$

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  • [JEE MAIN 2021]

Two identical balls having like charges and placed at a certain distance apart repel each other with a certain force. They are brought in contact and then moved apart to a distance equal to half their initial separation. The force of repulsion between them increases $4.5$ times in comparison with the initial value. The ratio of the initial charges of the balls is