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Question #42924

Difficulty: 8/10 | 🟢 Free

The Question

A solid conducting sphere of radius $R$ carrying a charge $Q$ is fixed in a gravity-free region. A given volume of a non-conducting, incompressible liquid of density $\rho$ and permittivity $\epsilon = 3\epsilon_0$ surrounds the sphere. The entire system is kept inside a large closed cabin maintained at a high constant gas pressure $P_{ext}$ . In the unaccelerated state, the liquid forms a spherical shell of outer radius $r_0$ around the conductor. The cabin is then given a uniform acceleration $a$ in the $+x$ direction. In the steady state, the liquid surface is slightly deformed. Let $r_1$ and $r_2$ be the distances from the center of the sphere to the liquid surface along the $+x$ and $-x$ axes, respectively. Assuming the deformation is small ( $|r_1 - r_0| \ll r_0$ ) and neglecting the effect of the liquid's non-spherical shape on the electric field distribution, which of the following expressions gives the difference $(r_2 - r_1)$ ?

Options

A.
$\frac{24 \pi^2 \epsilon_0 \rho a r_0^6}{Q^2}$
B.
$\frac{16 \pi^2 \epsilon_0 \rho a r_0^6}{Q^2}$
C.
$\frac{48 \pi^2 \epsilon_0 \rho a r_0^6}{Q^2}$
D.
$\frac{12 \pi^2 \epsilon_0 \rho a r_0^6}{Q^2}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Charged Conducting Sphere|Accelerated Frame Hydrostatics|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the superposition of fictitious force-induced hydrostatic pressure gradients and non-linear electrostatic Maxwell stress at a dielectric boundary.

A common trap is miscalculating the electrostatic pressure by either ignoring the dielectric polarization (using only the vacuum field, leading to Option B) or using only the internal field energy density (leading to Option C).

Furthermore, students often fail to correctly linearize the $1/r^4$ electrostatic pressure term using the binomial approximation to reconcile it with the linear hydrostatic pressure, or they misapply the boundary condition $P_{liq} + P_{es} = P_{ext}$ by getting the direction of the electrostatic pull wrong, which would lead to a sign error or confusion regarding which side of the sphere has a thicker liquid layer.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Charged Conducting Sphere
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free