Question

A charged particle is moving in a circular orbit with radius $r$ and orbital angular frequency $\omega$ in the presence of a magnetic field. The orbit is enclosed within a larger circular metallic frame. The frame is concentric and coplanar with the orbit. The radius of the frame is now gradually decreased. Assuming that the particle remains within the frame at all times, what changes to the trajectory of the particle will occur as the frame is being shrunk?

• A. The radius of the orbit will gradually decrease and the frequency will gradually increase.
• B. The radius of the orbit will gradually increase and the frequency will gradually decrease.
• C. The radius of the orbit will remain the same but the frequency will gradually increase.
• D. Both the radius of the orbit and the frequency will remain unchanged.

Hint:

Shrinking a conductor in a magnetic field always concentrates the flux lines inside it due to Lenz's law.
An increased $B$-field naturally means a smaller orbit radius ($r \propto B^{-1/2}$) and a higher orbital frequency ($\omega \propto B$).
This physical intuition lets you quickly choose Option (A) without deep calculations.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question explores electromagnetic induction in a shrinking conducting loop (frame) placed in a magnetic field, and the subsequent effect of the altered local magnetic field on a charged particle's cyclotron orbit.
Step 2: Key Formulas and Approach:
1. Lenz's Law and Faraday's Law:
As the conducting frame's area decreases, the magnetic flux through it decreases. This induces a current in the frame that opposes the change, increasing the magnetic field inside the frame.
2. Cyclotron frequency:
\[ \omega = \frac{q B}{m} \]
3. Adiabatic invariance of magnetic flux through the orbit:
For slowly changing magnetic fields, the magnetic flux through the particle's orbit is conserved:
\[ \Phi_{\text{orbit}} = B \cdot \pi r^2 = \text{constant} \]
Step 3: Detailed Explanation:

When the metallic frame is shrunk, the cross-sectional area enclosed by the frame decreases.

Since there is a magnetic field passing through the frame, this reduction in area leads to a decrease in the magnetic flux through the frame.

According to Faraday's Law and Lenz's Law, this changing flux induces an electromotive force (EMF) and a current in the conducting metallic frame.

The direction of the induced current is such that it generates its own magnetic field to oppose the reduction of the original flux. Thus, the induced magnetic field is in the same direction as the external magnetic field.

Consequently, the net magnetic field $B$ in the region enclosed by the frame increases.

The orbital angular frequency $\omega$ of the charged particle is given by:
\[ \omega = \frac{q B}{m} \]
Since $B$ increases, the frequency $\omega$ must gradually increase.

For a slowly changing magnetic field (adiabatic variation), the flux linked with the circular orbit of the particle remains invariant:
\[ B \cdot (\pi r^2) = \text{constant} \]

Because the magnetic field $B$ is increasing, the orbital radius $r$ must decrease to keep the product $B r^2$ constant.

Step 4: Final Answer:
The radius of the orbit will gradually decrease and the frequency will gradually increase, which matches Option (A).