Question

Which of the following Maxwell's equations is valid for time varying conditions but not valid for static conditions:

• A. \( \oint \vec{B} \cdot d\vec{l} = \mu_0 I \)
• B. \( \oint \vec{E} \cdot d\vec{l} = 0 \)
• C. \( \oint \vec{E} \cdot d\vec{l} = - \frac{\partial \vec{B}}{\partial t} \)
• D. \( \oint \vec{D} \cdot d\vec{A} = Q \)

Hint:

Faraday's Law of Induction is a cornerstone of electromagnetism, describing how electric fields are generated by time-varying magnetic fields. It is not valid under static conditions.

Answer 1

The Correct Option is C

Step 1: Maxwell's equations for time-varying conditions: One of the key equations that is valid only for time-varying conditions is Faraday's Law of Induction. It states that: \[ \oint \vec{E} \cdot d\vec{l} = - \frac{\partial \vec{B}}{\partial t} \] This equation describes how a changing magnetic field induces an electric field. 
Step 2: Understanding the options: - Option (A): This is Ampère's Law for magnetostatics, where the integral of \( \vec{B} \) around a closed loop is equal to the current enclosed. This equation is valid under static conditions as well. - Option (B): This is one form of the conservative electric field equation, valid for electrostatics, where the line integral of \( \vec{E} \) is zero for static conditions. - Option (C): This is the correct choice, as it describes Faraday's Law of Induction, which is valid only for time-varying conditions. - Option (D): This is Gauss's Law for electricity, valid for both static and dynamic conditions. 
Step 3: Verifying the correct answer. The equation \( \oint \vec{E} \cdot d\vec{l} = - \frac{\partial \vec{B}}{\partial t} \) is the correct choice as it applies only in time-varying situations (changing magnetic field).