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Question #43444

Difficulty: 8/10 | 🟢 Free

The Question

In a certain region of vacuum, an axially symmetric static magnetic field has a $z$ -component given by $B_z = B_0(1 + \alpha z)$ , where $B_0$ and $\alpha$ are positive constants. A bead of mass $m$ and charge $q > 0$ is constrained to slide without friction on the outer surface of a fixed non-conducting cylinder of radius $R_0$ whose axis coincides with the $z$ -axis. The bead is projected from the point $(R_0, 0, 0)$ with an initial velocity $v_0 \hat{k}$ . Assuming $v_0$ is sufficiently large so that the bead reaches the plane $z = d$ , what is the radial component of the normal force $N_r$ exerted by the cylinder on the bead at $z = d$ ? (Ignore gravity)

Options

A.
$\frac{q^2 B_0^2 R_0 \alpha d}{4m} (2 + \alpha d)$
B.
$\frac{q^2 B_0^2 R_0 \alpha d}{2m} (1 + \alpha d)$
C.
$\frac{q^2 B_0^2 R_0 \alpha d}{4m} (2 + 3\alpha d)$
D.
$0$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Gauss’s Law for Magnetism|Moving Charge in Non-Uniform B|Angular Momentum conservation

Why This Question Is Tricky

Common Mistakes

The core trap lies in the assumption that angular momentum is conserved ( $L_z = 0$ ) because the magnetic field is often incorrectly treated as purely axial ( $B_r = 0$ ).

Students failing to apply Gauss's Law for Magnetism ( $\nabla \cdot \mathbf{B} = 0$ ) will miss the induced radial magnetic field $B_r$ caused by the axial gradient $\frac{\partial B_z}{\partial z}$ .

This $B_r$ generates a tangential Lorentz force, inducing an angular velocity (a specific application of Busch's Theorem).

A secondary trap is neglecting the centripetal acceleration requirement in the radial force balance, leading to an incorrect normal force that only balances the radial Lorentz force.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Gauss’s Law for Magnetism
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free