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Question #42724

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical container of length $L$ (in m) and cross-sectional area $A$ (in m $^2$ ) is filled with a liquid metal. The uncompressed mass density of the liquid is $\rho_m$ (in kg/m $^3$ ), its bulk modulus is $K$ (in Pa), and its uncompressed electrical conductivity is $\sigma_0$ (in $\Omega^{-1}$ m $^{-1}$ ). The container is accelerated uniformly with acceleration $a$ (in m/s $^2$ ) along its central axis (the $+z$ direction, from $z=0$ to $z=L$ ). A steady electrical current $I$ (in A) flows through the liquid metal in the $+z$ direction. Assume the mean free time $\tau$ (in s) of the conduction electrons remains constant, and the local conductivity is strictly proportional to the local mass density. The mass of an electron is $m_e$ (in kg) and its charge is $-e$ (in C). The pressure at the front face ( $z=L$ ) is maintained at the uncompressed pressure $P_0$ (in Pa). Assuming $\rho_m a L \ll K$ , calculate the total flux of the Poynting vector (in W) into the lateral surface of the liquid metal cylinder. Let $R_0 = \frac{L}{\sigma_0 A}$ be the unaccelerated resistance of the cylinder.

Options

A.
$I^2 R_0 \left( 1 - \frac{\rho_m a L}{2 K} \right) - \frac{m_e a L I}{e}$
B.
$I^2 R_0 \left( 1 - \frac{\rho_m a L}{2 K} \right) + \frac{m_e a L I}{e}$
C.
$I^2 R_0 \left( 1 + \frac{\rho_m a L}{2 K} \right) - \frac{m_e a L I}{e}$
D.
$I^2 R_0 \left( 1 - \frac{\rho_m a L}{K} \right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Current Density Vector|Poynting Vector and Intensity|Accelerated Frame Hydrostatics|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This problem tests the synthesis of macroscopic fluid mechanics and microscopic electrodynamics in non-inertial frames.

Students often fall into two traps: first, they assume a 'steady current' implies a uniform electric field, ignoring that the hydrostatic pressure gradient in the accelerated frame compresses the liquid metal, locally altering the carrier density and conductivity.

Second, they neglect the Tolman-Stewart effect, where the inertial pseudo-force acts directly on the charge carriers.

Because the current is in the $+z$ direction, electrons drift in the $-z$ direction; the $-z$ pseudo-force assists this drift, reducing the external electric field required to maintain the current.

The Poynting vector flux, which represents the electromagnetic power entering the conductor, is strictly the integral of this modified, position-dependent electric field.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Current Density Vector
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free