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Question #42093

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ and charge $q > 0$ is moving in a circular orbit of radius $r_0$ with a constant speed $v_0$ in the $xy$ -plane. For $t < 0$ , the region contains a uniform and constant magnetic field $\vec{B} = B_0 \hat{k}$ . At $t = 0$ , the magnetic field begins to increase linearly with time as $\vec{B}(t) = B_0(1 + \alpha t) \hat{k}$ , where $\alpha > 0$ is a small constant. Find the magnitude of the radial component of the jerk vector (defined as the rate of change of the acceleration vector, $\vec{j} = \frac{d\vec{a}}{dt}$ ) of the particle exactly at $t = 0^+$ . Assume non-relativistic motion.

Options

A.
$2 \alpha v_0 \left( \frac{qB_0}{m} \right)$
B.
$\frac{3}{2} \alpha v_0 \left( \frac{qB_0}{m} \right)$
C.
$\frac{1}{2} \alpha v_0 \left( \frac{qB_0}{m} \right)$
D.
$\alpha v_0 \left( \frac{qB_0}{m} \right)$

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

Centripetal Acceleration Vector|Cyclotron Frequency Invariance|Integration

Why This Question Is Tricky

Common Mistakes

This problem tests the coupling of kinematics and electromagnetism in a time-varying magnetic field.

Students often fall into the trap of either ignoring the induced electric field (Faraday's Law) or incorrectly differentiating the acceleration vector in polar coordinates.

If the induced electric field is ignored, the tangential acceleration is zero, leading to an incorrect radial jerk of $\alpha v_0 (qB_0/m)$ .

If the student correctly finds the induced electric field but forgets the coordinate frame rotation term $-a_\theta \dot{\theta}$ in the radial jerk component, they obtain the distractor $\frac{3}{2} \alpha v_0 (qB_0/m)$ .

The correct solution requires applying the exact radial equation of motion, differentiating it to find $\dddot{r}$ , and properly evaluating the full polar coordinate expression for the jerk vector.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Centripetal Acceleration Vector
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free