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Question #43386

Difficulty: 8/10 | 🟢 Free

The Question

In an isochronous cyclotron, the magnetic field $B(r)$ is designed to increase with the radial distance $r$ from the center to exactly compensate for the relativistic mass increase of the accelerated particles. This ensures that the orbital angular frequency $\omega_0$ remains strictly constant. A proton (rest mass $m_0$ , charge $q$ ) is accelerated from rest at the center, where the magnetic field is $B_0$ . Let $c$ be the speed of light in vacuum. What is the radial gradient of the magnetic field, $\frac{dB}{dr}$ , at the specific orbit where the proton reaches a speed $v = \frac{\sqrt{3}}{2}c$ ?

Options

A.
$4\sqrt{3} \frac{q B_0^2}{m_0 c}$
B.
$2\sqrt{3} \frac{q B_0^2}{m_0 c}$
C.
$\sqrt{3} \frac{q B_0^2}{m_0 c}$
D.
$8\sqrt{3} \frac{q B_0^2}{m_0 c}$

Check Your Answer

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

Cyclotron Frequency Invariance|Orbital Velocity Energy Balance|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate classical cyclotron mechanics with special relativity.

The primary trap lies in the assumption that the cyclotron frequency is universally constant without requiring a spatially varying magnetic field.

Students often misapply the relativistic mass correction, either by incorrectly differentiating the Lorentz factor $\gamma$ with respect to radius (forgetting the chain rule linking $v$ and $r$ via $\omega_0$ ), or by dropping the $\gamma^3$ term that emerges from the derivative of $(1-v^2/c^2)^{-1/2}$ .

This highlights the engineering necessity of the isochronous cyclotron to prevent beam dephasing at relativistic speeds.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Cyclotron Frequency Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free