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Question #43503

Difficulty: 8/10 | 🟢 Free

The Question

A conducting rod of mass $m$ and length $l$ is placed on two long, frictionless, horizontal conducting rails. A uniform vertical magnetic field $B$ exists throughout the region. The rails are connected at one end by a non-ohmic electronic device. The dynamic resistance of this device is defined as $R_d = \frac{dV}{dI} = \alpha I$ , where $V$ is the voltage drop across the device, $I$ is the current through it ( $I \ge 0$ ), and $\alpha$ is a positive constant. Assume $V=0$ when $I=0$ . The rod is given an initial velocity $v_0$ parallel to the rails. Neglecting the electrical resistance of the rod and the rails, and ignoring any self-inductance in the circuit, what is the total distance $x_f$ the rod travels before coming to rest?

Options

A.
$\frac{\sqrt{2} m \sqrt{\alpha} v_0^{3/2}}{3 (B l)^{3/2}}$
B.
$\frac{2 m \sqrt{\alpha} v_0^{3/2}}{3 (B l)^{3/2}}$
C.
$\frac{m \sqrt{\alpha} v_0^{3/2}}{3 (B l)^{3/2}}$
D.
$\frac{2\sqrt{2} m \sqrt{\alpha} v_0^{3/2}}{3 (B l)^{3/2}}$

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

Power-Velocity Relationship|Dynamic Resistance (AC Resistance)|Cell EMF vs Terminal Voltage

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply the concept of dynamic resistance in a coupled electromechanical system.

The lethal trap lies in the misapplication of Ohm's Law for power or voltage in non-ohmic devices.

Students commonly assume that voltage is simply $V = I R_d = \alpha I^2$ , or that power dissipated is $P = I^2 R_d = \alpha I^3$ .

Both assumptions fail because dynamic resistance is a differential quantity ( $R_d = dV/dI$ ).

The correct macroscopic voltage must be found via integration ( $V = \int R_d dI = \frac{1}{2}\alpha I^2$ ).

Failing to integrate introduces a missing factor of $1/2$ in the voltage-current relationship, leading directly to the incorrect coefficient in Option B.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Power-Velocity Relationship
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free