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Question #43042

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass mm and charge qq is projected with a velocity v⃗=v0j^\vec{v} = v_0 \hat{j} from the position r⃗=x0i^\vec{r} = x_0 \hat{i} in an inertial laboratory frame SS. A non-uniform magnetic field B⃗=B0(1+αx)k^\vec{B} = B_0(1 + \alpha x) \hat{k} exists in the region, where B0B_0 and α\alpha are positive constants. A non-inertial frame S′S' rotates with a constant angular velocity ω⃗=ωk^\vec{\omega} = \omega \hat{k} relative to SS. Assume the coordinate axes of SS and S′S' perfectly coincide at the instant of projection (t=0t=0). What is the initial acceleration a⃗′\vec{a}' of the particle as measured by an observer in the rotating frame S′S' at t=0t=0?

Options

  • A.

    [qv0B0(1+αx0)m+2ωv0−ω2x0]i^\left[ \frac{q v_0 B_0(1 + \alpha x_0)}{m} + 2\omega v_0 - \omega^2 x_0 \right] \hat{i}

  • B.

    [q(v0−ωx0)B0(1+αx0)m+2ωv0−ω2x0]i^\left[ \frac{q (v_0 - \omega x_0) B_0(1 + \alpha x_0)}{m} + 2\omega v_0 - \omega^2 x_0 \right] \hat{i}

  • C.

    [qv0B0(1+αx0)m+ω2x0]i^\left[ \frac{q v_0 B_0(1 + \alpha x_0)}{m} + \omega^2 x_0 \right] \hat{i}

  • D.

    [q(v0−ωx0)B0(1+αx0)m]i^\left[ \frac{q (v_0 - \omega x_0) B_0(1 + \alpha x_0)}{m} \right] \hat{i}

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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

Moving Charge in Non-Uniform B|Frame of Reference Dependency|Center of Mass (COM) Frame

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply Newton's Second Law in a rotating non-inertial frame, specifically focusing on the transformation of the Lorentz force.

A common lethal mistake (Trap B or D) is assuming the magnetic force in the rotating frame is simply q(v⃗′×B⃗)q(\vec{v}' \times \vec{B}), failing to account for the induced motional electric field E⃗′=(ω⃗×r⃗′)×B⃗\vec{E}' = (\vec{\omega} \times \vec{r}') \times \vec{B} that appears in the rotating frame.

Alternatively, students may correctly use the lab-frame real force but forget to add the fictitious Centrifugal and Coriolis forces (Trap C).

The problem rigorously demands the synthesis of Galilean velocity transformations, fictitious forces, and the frame-invariance of the total real Lorentz force.

Quick Facts About This Question

Question Information

SubjectPhysics
ChapterMoving Charges and Magnetism - Class 12
ConceptMoving Charge in Non-Uniform B
Difficulty8
Cognitive Load7
Target Solve Time160 sec
AccessFree