Before You Begin

Question #42970

Difficulty: 8/10 | 🟢 Free

The Question

A conducting rod of mass $m$ (in kg) and length $L$ (in m) slides down a rough inclined plane of angle $\theta$ with the horizontal. The plane has a coefficient of kinetic friction $\mu$ . The rod is in contact with two parallel, perfectly conducting rails fixed to the incline, which are connected at the bottom by a resistor of resistance $R$ (in $\Omega$ ). A uniform magnetic field of magnetic induction $B$ (in T) exists vertically downwards throughout the region. The rod is released from rest. Assuming the rails are very long, what is the terminal velocity $v_0$ (in m/s) acquired by the rod? (Neglect the self-inductance of the loop and electrical resistance of the rod and rails).

Options

A.
$\frac{mg R (\sin\theta - \mu \cos\theta)}{B^2 L^2 \cos\theta (\cos\theta + \mu \sin\theta)}$
B.
$\frac{mg R (\sin\theta - \mu \cos\theta)}{B^2 L^2 \cos^2\theta}$
C.
$\frac{mg R (\sin\theta - \mu \cos\theta)}{B^2 L^2}$
D.
$\frac{mg R (\sin\theta - \mu \cos\theta)}{B^2 L^2 \cos\theta (\cos\theta - \mu \sin\theta)}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Magnetic Flux Scalar Sign|Eddy Currents Dissipation|Wedge Constraint Logic

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of electromagnetic induction with non-conservative mechanical forces via a modified normal force constraint.

The primary trap is the assumption that the normal force on an incline remains $mg \cos\theta$ .

Students often correctly identify the horizontal magnetic braking force $F_m$ but fail to resolve its component perpendicular to the incline.

Because $F_m$ is horizontal and opposes the horizontal component of velocity, it pushes the rod into the wedge, increasing the normal force to $N = mg \cos\theta + F_m \sin\theta$ .

This creates a secondary, velocity-dependent braking effect through increased kinetic friction, fundamentally altering the terminal velocity equation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Magnetic Flux Scalar Sign
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free