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Question #43760

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The Question

A uniform magnetic field $\vec{B} = B_0(1 + \beta t) \hat{k}$ exists in a cylindrical region of radius $R$ whose axis coincides with the $z$ -axis. A particle of mass $m$ and positive charge $q$ is located at a point $(r_0, 0, 0)$ at time $t = 0$ , where $r_0 < R$ . At this instant, the particle is projected with an initial velocity $\vec{v} = v_0 \hat{i}$ . Assuming $v_0 > \frac{r_0 \beta}{2}$ , what is the instantaneous radius of curvature of the particle's trajectory at $t = 0$ ?

Options

A.
$\frac{m v_0^2}{q B_0 \left( v_0 + \frac{r_0 \beta}{2} \right)}$
B.
$\frac{m v_0}{q B_0}$
C.
$\frac{m v_0^2}{q B_0 \left( v_0 - \frac{r_0 \beta}{2} \right)}$
D.
$\frac{2 m v_0^2}{q B_0 r_0 \beta}$

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

Instantaneous Nature|Lorentz Force Vector Direction|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the instantaneous nature of the Lorentz force in a time-varying magnetic field.

The primary trap is the assumption that the radius of curvature is simply the cyclotron radius $\rho = mv/qB$ , which only holds for static magnetic fields where no electric field is present.

By introducing $dB/dt \neq 0$ , Maxwell-Faraday's law mandates an induced non-conservative electric field.

Students who fail to account for the electric force $\vec{F}_e = q\vec{E}$ will incorrectly choose the standard cyclotron formula.

Furthermore, the problem tests the correct application of Lenz's law to determine the direction of the induced electric field; a sign error here leads to an incorrect subtraction of forces.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Dual Nature of Radiation and Matter - Class 12
Concept	Instantaneous Nature
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free