Answer: The equation shown is Faraday’s Law of Electromagnetic Induction in differential form.
Explanation:
This equation is a fundamental Maxwell’s equation describing how a time-varying magnetic field induces an electric field. Specifically, it states that the curl of the electric field \(\nabla \times \vec{E}\) is equal to the negative rate of change of the magnetic flux density \(\vec{B}\) with respect to time. The theorem involved here is Maxwell’s Equations, particularly Faraday’s Law in differential form.
Steps:
- Recognize the notation:
- \(\nabla \times \vec{E}\) is the curl of the electric field, representing the rotation or circulation of \(\vec{E}\).
- \(\frac{\partial \vec{B}}{\partial t}\) is the partial derivative of the magnetic flux density \(\vec{B}\) with respect to time.
- Recall the physical principle:
- A changing magnetic field induces an electric field, which is described mathematically by Faraday’s Law.
- The differential form of Faraday’s Law is:
\[
\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}
\]
- The formula relates the spatial variation (curl) of \(\vec{E}\) to the temporal variation of \(\vec{B}\).
Therefore, the given equation is directly derived from Maxwell’s equations, specifically Faraday’s Law of Induction in differential form.