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Question #43387

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The Question

A particle of charge $q$ and mass $m$ is accelerated in a betatron, maintaining a circular orbit of constant radius $R$ . The magnetic field exactly at the location of the orbit varies with time $t$ as $B(t) = B_0 t$ , where $B_0$ is a positive constant. Assuming non-relativistic mechanics and that the particle starts from rest at $t=0$ , which of the following expressions correctly gives the required rate of change of magnetic flux $\frac{d\Phi}{dt}$ enclosed by the orbit, and the instantaneous power $P$ delivered to the particle at time $t$ ?

Options

A.
$\frac{d\Phi}{dt} = 2\pi R^2 B_0 \quad \text{and} \quad P = \frac{q^2 B_0^2 R^2 t}{m}$
B.
$\frac{d\Phi}{dt} = \pi R^2 B_0 \quad \text{and} \quad P = \frac{q^2 B_0^2 R^2 t}{4m}$
C.
$\frac{d\Phi}{dt} = 2\pi R^2 B_0 \quad \text{and} \quad P = \frac{q^2 B_0^2 R^2 t}{2m}$
D.
$\frac{d\Phi}{dt} = \pi R^2 B_0 \quad \text{and} \quad P = \frac{q^2 B_0^2 R^2 t}{m}$

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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

Cyclotron Frequency Invariance|Induced Electric Fields|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the understanding of Widerøe's condition in a betatron and traps students who incorrectly assume the magnetic field is spatially uniform.

A common mistake is to write $\Phi = \pi R^2 B(t)$ , leading to $\frac{d\Phi}{dt} = \pi R^2 B_0$ .

However, to maintain a constant radius while the particle's momentum increases, the induced electric field must provide a specific tangential acceleration.

By applying Newton's Second Law and Faraday's Law, one finds that the average magnetic field enclosed by the orbit must be exactly twice the magnetic field at the orbit itself.

This requires students to derive the flux rate dynamically from the required momentum change rather than from a flawed geometric assumption.

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