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Question #42761

Difficulty: 8/10 | 🟢 Free

The Question

A solid, uncharged conducting sphere of radius $R$ is placed inside a rocket that is accelerating in deep space with a constant acceleration $\vec{a} = a\hat{i}$ . Let the center of the sphere be at the origin $O(0,0,0)$ . Let $m_e$ and $-e$ be the mass and charge of an electron, respectively. Assume that the positive ions of the conductor's lattice are fixed relative to the sphere. Once electrostatic equilibrium is reached in the frame of the rocket, what is the potential difference $V_A - V_O$ between the point $A(R, 0, 0)$ on the surface of the sphere and the center $O$ , and what is the induced surface charge density $\sigma_A$ at point $A$ ?

Options

A.
$V_A - V_O = \frac{m_e a R}{e}, \quad \sigma_A = \frac{3 \epsilon_0 m_e a}{e}$
B.
$V_A - V_O = \frac{m_e a R}{e}, \quad \sigma_A = \frac{\epsilon_0 m_e a}{e}$
C.
$V_A - V_O = 0, \quad \sigma_A = 0$
D.
$V_A - V_O = -\frac{m_e a R}{e}, \quad \sigma_A = -\frac{3 \epsilon_0 m_e a}{e}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Conductor Interior Potential|Frame of Reference Inertial vs Non-Inertial|Pseudo Force Directionality

Why This Question Is Tricky

Common Mistakes

This question tests the limits of the 'equipotential volume' mnemonic for conductors.

Students are heavily conditioned to assume that the electric field inside a conductor in electrostatic equilibrium is always zero ( $E=0$ ), which would lead them to incorrectly choose Option C.

However, this rule only applies in an inertial frame without non-electrical body forces.

In an accelerating frame, the pseudo-force acts as an effective uniform gravitational field on the electron gas.

To maintain equilibrium, the electrons must redistribute to create a non-zero internal electric field that exactly cancels the pseudo-force.

A secondary trap (Option B) tests whether students correctly apply the boundary conditions for a polarized sphere (where $E_{in} = \sigma_0 / 3\epsilon_0$ ) rather than incorrectly applying the infinite slab formula ( $E_{in} = \sigma / \epsilon_0$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Conductor Interior Potential
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free