Two identical charged spheres are suspended by string of equal lengths. The string make an angle of $37^{\circ}$ with each other. When suspended in a liquid of density $0.7\mathrm{~g}/\mathrm{cm}^{3}$, the angle remains same. If density of material of the sphere is $1.4\mathrm{~g}/\mathrm{cm}^{3}$, the dielectric constant of the liquid is ____ $(\tan 37^{\circ}=\frac{3}{4})$.

JEE Main · 2024 · 30 Jan 2024 · Shift 2 · Electrostatics

Two identical charged spheres are suspended by string of equal lengths. The string make an angle of

3 7^{\circ}

with each other. When suspended in a liquid of density

0.7 g / cm^{3}

, the angle remains same. If density of material of the sphere is

1.4 g / cm^{3}

, the dielectric constant of the liquid is ____

(tan 3 7^{\circ} = \frac{3}{4})

Answer: 2

In air, the string tension, gravity and the electrostatic repulsion between the identical charged spheres balance:

T cos θ = m g, T sin θ = F_{e}

tan θ = \frac{F _{e}}{m g} = \frac{F _{e}}{ρ _{B} V g}

When the spheres are suspended in a liquid of density

ρ_{L}

and dielectric constant

k

, the electrostatic force is reduced by a factor

k

, and gravity is effectively reduced by buoyancy:

tan θ = \frac{F _{e} / k}{( ρ _{B} - ρ _{L} ) V g}

Since the angle

θ

(and hence

tan θ

) is unchanged between the two cases:

\frac{F _{e}}{ρ _{B} V g} = \frac{F _{e} / k}{( ρ _{B} - ρ _{L} ) V g}

ρ_{B} = k (ρ_{B} - ρ_{L})

Substituting

ρ_{B} = 1.4 g/cm^{3}

and

ρ_{L} = 0.7 g/cm^{3}

1.4 = k (1.4 - 0.7) = 0.7 k

k = 2

JEE Main · 2024 · 30 Jan 2024 · Shift 2 · Integer answer