Question

A charged particle of charge q and mass m is placed at a distance 2R from the center of a vertical cylindrical region of radius R where the magnetic field varies as \(\vec{B}=(4t^2-2t+6)k\), where t is time. Then which of the following statement(s) is/are true?

• A. Induced electric field lines from closed loops
• B. The Electric field varies linearly with r if r< R, where r is the radial distance from the centerline of the cylinder.
• C. The charged particle will move in a clockwise direction when viewed from the top.
• D. Acceleration of the charged particle is \(\frac{7q}{2m}\) when t=2 sec.

Answer 1

The Correct Option is A, B, C

The correct answer is/are option(s):
(A): Induced electric field lines from closed loops
(B): The Electric field varies linearly with r if r< R, where r is the radial distance from the centerline of the cylinder.
(C): The charged particle will move in a clockwise direction when viewed from the top.

Answer 2

Given a charged particle of charge \( q \) and mass \( m \) placed at a distance \( 2R \) from the center of a vertical cylindrical region of radius \( R \), with a time-varying magnetic field \( B = (4t^2 - 2t + 6) \hat{k} \), we need to determine which of the provided statements are true.
Magnetic Field and Induced Electric Field
The magnetic field \( B = (4t^2 - 2t + 6) \hat{k} \) varies with time, which according to Faraday's law of electromagnetic induction, will induce an electric field.
Faraday's Law of Induction
Faraday's law states:
\[ \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \]
Using the integral form for a circular loop of radius \( r \) centered on the axis of the cylinder:
\[ \oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt} \]
where \( \Phi_B \) is the magnetic flux through the loop:
\[ \Phi_B = \int_S \vec{B} \cdot d\vec{A} = B \cdot \pi r^2 \]
Given \( B = (4t^2 - 2t + 6) \hat{k} \):
\[ \Phi_B = (4t^2 - 2t + 6) \cdot \pi r^2 \]
The induced emf is:
\[ \mathcal{E} = -\frac{d\Phi_B}{dt} = -\pi r^2 \frac{d}{dt} (4t^2 - 2t + 6) \]
\[ \frac{d}{dt} (4t^2 - 2t + 6) = 8t - 2 \]
\[ \mathcal{E} = -\pi r^2 (8t - 2) \]
The induced electric field \( \vec{E} \) is tangential to the loop and has magnitude:
\[ \mathcal{E} = E \cdot 2\pi r \]
\[ E \cdot 2\pi r = -\pi r^2 (8t - 2) \]
\[ E = -\frac{r}{2} (8t - 2) \]
Analysis of Statements
(A) Induced electric field lines form closed loops:
  - According to Faraday's law, the induced electric field forms closed loops perpendicular to the changing magnetic flux. This statement is true.
(B) The Electric field varies linearly with \( r \) if \( r < R \):
  - From the derived expression for \( E \), it is clear that \( E \) varies linearly with \( r \). This statement is true.
(C) The charged particle will move in a clockwise direction when viewed from the top:
  - The direction of the induced electric field will determine the direction of the force on the charged particle. According to Lenz's law, the induced current (and hence the motion of a positive charge) will oppose the change in magnetic flux. Given that the magnetic field is increasing, the induced electric field will create a force that opposes this increase. For a positively charged particle, this will result in a clockwise motion when viewed from above. This statement is true.
 Conclusion
The correct statements are:
(A) Induced electric field lines form closed loops.
(B) The Electric field varies linearly with \( r \) if \( r < R \).
(C) The charged particle will move in a clockwise direction when viewed from the top.