Before You Begin

Question #42813

Difficulty: 8/10 | 🟢 Free

The Question

A straight conducting rod of length $L$ has the shape of a truncated cone with circular end faces of radii $a$ and $b$ ( $b > a$ ). The rod is made of an isotropic material of uniform electrical resistivity $\rho$ . A constant potential difference $V$ is applied across its end faces. Assuming that the current density vector is uniform across any cross-section and parallel to the axis of the rod, which of the following expressions gives the magnitude of the electric field at the exact midpoint of the rod's axis?

Options

A.
$\frac{4 V a b}{L (a+b)^2}$
B.
$\frac{V}{L}$
C.
$\frac{V (a^2 + b^2)}{2 L a b}$
D.
$\frac{2 V (a^2 + b^2)}{L (a+b)^2}$

Check Your Answer

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

Drift Velocity Mechanism|Potential Gradient Direction|Steady State Temperature Gradient

Why This Question Is Tricky

Common Mistakes

This question targets the common misconception that the electric field inside a current-carrying conductor is always uniform ( $E = V/L$ ).

Students often misapply this to non-uniform geometries, ignoring the continuity of steady current ( $I = J \cdot A = \text{constant}$ ).

Because $J = E/\rho$ , the electric field must vary inversely with the cross-sectional area ( $E \propto 1/A$ ).

By forcing the derivation of the spatial variation of $E(x)$ , the problem exposes the flaw of assuming linear potential drops or using average areas, highlighting that non-uniform conductors require surface charge distributions to maintain the steady-state current field.

Quick Facts About This Question

Question Information

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