Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is . The spheres are now immersed in a dielec

Question

Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is $α$ . The spheres are now immersed in a dielectric liquid of density 800 kg $m^{- 3}$ and dielectric constant 21. If the angle between the strings remains the same after the immersion, then [2020]

Smart Achievers · Accepted Answer

(1, 3)

The net electric force on any sphere is lesser but by Coulomb's law, the force due to one sphere to another remains the same.

In equilibrium

$T \cos θ = m g$

and $T \sin θ = F$

$∴$ $\tan θ = \frac{F}{m g}$ ...(i)

As force between two charged bodies according to Coulomb's law does not depend upon the medium, hence force between them remains same.

Hence after immersed in dielectric medium, as given no change in angle $θ$ .

$T' \sin θ = \frac{F}{ε_{r}}$ and $T' \cos θ = m g (1 - \frac{ρ_{l}}{ρ_{s}})$

$∴$ $\tan θ = \frac{F}{m g (1 - \frac{ρ_{l}}{ρ_{s}})} \cdot \frac{1}{ε_{r}}$ ...(ii)

Now equating Eqns. (i) and (ii),

$1 - \frac{ρ_{l}}{ρ_{s}} = \frac{1}{ε_{r}}$ $\Rightarrow 1 - \frac{800}{ρ_{s}} = \frac{1}{21}$

$∴$ $ρ_{s} = 840 {kg/m}^{3}$

Hence option (3) is correct.

$And tension, T' = \frac{T}{ε_{r}} = \frac{T}{21}$

Hence option (4) is incorrect.

Solution

1 electric force between the spheres remains unchanged

2 electric force between the spheres reduces

3 mass density of the spheres is 840 kg $m^{- 3}$

4 the tension in the strings holding the spheres remains unchanged

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