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Question #42456

Difficulty: 8/10 | 🟢 Free

The Question

A small block of mass $m$ and positive charge $q$ is projected up a rough inclined plane with an initial velocity $v_0$ . The angle of inclination of the plane is $\theta$ and the coefficient of kinetic friction between the block and the plane is $\mu$ . A uniform magnetic field of magnitude $B$ exists in the region. The magnetic field is directed parallel to the incline surface and perpendicular to the line of greatest slope, such that the resulting magnetic force pushes the block perpendicularly into the incline. Acceleration due to gravity is $g$ . Which of the following expressions correctly represents the maximum distance $d$ traveled by the block up the incline before coming to rest momentarily?

Options

A.
$\frac{m v_0}{\mu q B} - \frac{m^2 g(\sin\theta + \mu \cos\theta)}{\mu^2 q^2 B^2} \ln\left( 1 + \frac{\mu q B v_0}{m g(\sin\theta + \mu \cos\theta)} \right)$
B.
$\frac{v_0^2}{2g(\sin\theta + \mu \cos\theta)}$
C.
$\frac{m v_0}{q B} - \frac{m^2 g(\sin\theta + \mu \cos\theta)}{q^2 B^2} \ln\left( 1 + \frac{q B v_0}{m g(\sin\theta + \mu \cos\theta)} \right)$
D.
$\frac{m v_0}{\mu q B} - \frac{m^2 g(\sin\theta - \mu \cos\theta)}{\mu^2 q^2 B^2} \ln\left( 1 + \frac{\mu q B v_0}{m g(\sin\theta - \mu \cos\theta)} \right)$

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

Minimum Force to Move a Block|Lorentz Force Vector Direction|Trajectory of Charged Particle

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle a 'Coupled Variable Constraint' in mechanics.

A common misconception is treating the normal force as a static value ( $mg \cos\theta$ ) when a charged particle moves in a magnetic field.

Here, the Lorentz force $\vec{F}_m = q(\vec{v} \times \vec{B})$ acts perpendicularly into the incline, dynamically increasing the normal force as a function of velocity.

Consequently, the kinetic friction becomes velocity-dependent ( $f_k = \mu(mg \cos\theta + qvB)$ ).

Students who fail to recognize this will incorrectly use constant acceleration equations.

Furthermore, the problem requires setting up and solving a non-linear differential equation with respect to position ( $v \frac{dv}{dx}$ ), testing mathematical rigor in integrating rational functions.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Minimum Force to Move a Block
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free