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Question #43358

Difficulty: 8/10 | 🟢 Free

The Question

An electrostatic potentiometer is constructed using a long cylindrical capacitor of inner radius $a$ and outer radius $b$ . The region between the cylinders ( $a \le r \le b$ ) is filled with a non-uniform dielectric medium whose relative permittivity varies with radial distance $r$ as $\kappa(r) = \kappa_0 \frac{r}{a}$ , where $\kappa_0$ is a positive constant. A constant DC voltage source maintains the inner cylinder at a potential $V_0$ and the outer cylinder at ground ( $0\text{ V}$ ). A radial probe is used to find a null point where the local potential balances an ideal cell of EMF $\mathcal{E} = \frac{V_0}{2}$ connected between the probe and the grounded outer cylinder. At what radial distance $r_0$ from the central axis will the null point be obtained?

Options

A.
$\frac{a+b}{2}$
B.
$\sqrt{ab}$
C.
$\frac{2ab}{a+b}$
D.
$\frac{a^2+b^2}{a+b}$

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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

Potentiometer Sensitivity|Gauss Law Applicability|Dielectric Polarization Logic

Why This Question Is Tricky

Common Mistakes

This question tests the misapplication of potential gradient linearity in non-uniform media and geometries.

Students often assume a linear potential drop $V(r) \propto r$ (yielding the arithmetic mean $\frac{a+b}{2}$ ) as in a standard wire potentiometer, completely ignoring the cylindrical geometry.

A secondary trap is assuming a standard logarithmic potential drop $V(r) \propto \ln(r)$ (yielding the geometric mean $\sqrt{ab}$ ), which accounts for the geometry but ignores the spatially varying dielectric constant.

The correct approach requires deriving the local electric field from Gauss's Law for the displacement field ( $\nabla \cdot \vec{D} = \rho_{\text{free}}$ ) and integrating $E(r) = D(r)/\epsilon(r)$ to find the true non-linear potential distribution, which in this specific case yields the harmonic mean.

Quick Facts About This Question

Question Information

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