Validity of Electric and Magnetic Field Patterns

To determine which field pattern can represent both an electric field ($\vec{E}$) and a magnetic field ($\vec{B}$), we analyze the fundamental governing equations of electromagnetism (Maxwell's equations):

1. Gauss's Law for Magnetism:
   $\vec{\nabla} \cdot \vec{B} = 0$
   This equation states that isolated magnetic monopoles do not exist. As a result, magnetic field lines must be continuous and cannot originate from or terminate at a single point source. This rules out field patterns (A) and (B) for magnetic fields, as they depict field lines diverging from or converging to a point source.

2. Field Line Behavior for Electric Fields:

   • An electrostatic field is conservative ($\vec{\nabla} \times \vec{E} = 0$), where field lines originate on positive charges and terminate on negative charges.
   • An induced electric field is non-conservative ($\vec{\nabla} \times \vec{E} \neq 0$) and is generated by a time-varying magnetic flux according to Faraday's Law of Induction:
     $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$
     In a region with cylindrical symmetry and a time-varying magnetic field (e.g., inside a solenoid with a changing current), the induced electric field lines form closed, concentric circles.

3. Field Line Behavior for Magnetic Fields:

   • By Ampère's Circuital Law:
     $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enclosed}}$
     The magnetic field created by a long, straight, current-carrying wire perpendicular to a plane forms closed, concentric circular field lines centered on the conductor.

4. Conclusion for Pattern (C):

   • Pattern (C) shows closed concentric circular field lines.
   • It represents a valid magnetic field generated by a steady electric current flowing perpendicular to the plane.
   • It also represents a valid induced electric field created by a changing magnetic field directed perpendicular to the plane.

Therefore, pattern (C) is valid for both electric and magnetic fields.

Correct Answer: C