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Question #42686

Difficulty: 8/10 | 🟢 Free

The Question

An incompressible, non-viscous fluid of mass density $\rho$ ( $\text{kg/m}^3$ ), absolute permittivity $\epsilon$ ( $\text{F/m}$ ), and uniform free charge density $\rho_e$ ( $\text{C/m}^3$ ) flows steadily and irrotationally through a horizontal pipe. The pipe has a wide section of cross-sectional area $2A$ ( $\text{m}^2$ ) (Region 1) and a narrow section of area $A$ ( $\text{m}^2$ ) (Region 2). Region 1 is in a field-free space. Region 2 passes through a parallel-plate capacitor of plate separation $d$ ( $\text{m}$ ), which maintains a uniform transverse electric field $E$ ( $\text{V/m}$ ) perpendicular to the flow direction. The volume flow rate of the fluid is $Q$ ( $\text{m}^3\text{/s}$ ). Assuming the flow remains strictly one-dimensional and accounting for the electro-hydrodynamic energy balance, what is the pressure difference $P_1 - P_2$ ( $\text{Pa}$ ) between Region 1 and Region 2 along a central streamline?

Options

A.
$\frac{3\rho Q^2}{8A^2} + \frac{1}{2}\epsilon E^2$
B.
$\frac{3\rho Q^2}{8A^2}$
C.
$\frac{3\rho Q^2}{8A^2} - \frac{1}{2}\epsilon E^2$
D.
$\frac{3\rho Q^2}{8A^2} + \rho_e E d$

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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

Energy Density Locality|Bernoulli’s Energy Balance|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the understanding of electro-hydrodynamic energy coupling in fluid mechanics.

A common trap is for students to focus solely on the volumetric Lorentz force ( $\rho_e E$ ).

Since the electric field is transverse to the flow, the Lorentz force does no longitudinal work, leading students to erroneously apply the standard Bernoulli equation and ignore the electric field entirely (Option B), or incorrectly use the transverse work as a longitudinal pressure drop (Option D).

However, they fail to account for the electric energy density.

A transverse electric field generates a longitudinal Maxwell stress that acts as an effective pressure.

This energy density ( $\frac{1}{2}\epsilon E^2$ ) must be incorporated into the Bernoulli energy balance as an additional internal energy term, modifying the conservation equation to $P + \frac{1}{2}\rho v^2 + \frac{1}{2}\epsilon E^2 = \text{constant}$ .

Quick Facts About This Question

Question Information

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