What is the third Maxwell equation?
2 Answers
The third Maxwell equation is known as Faraday's law of induction. It states that a changing magnetic field within a closed loop induces an electromotive force (EMF) in that loop. Mathematically, it's expressed as:
\[
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
\]
This equation highlights the relationship between electric fields and changing magnetic fields, ultimately illustrating how electric currents can be generated through magnetic changes. If you want to delve deeper into its implications or applications, let me know!
\[
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
\]
This equation highlights the relationship between electric fields and changing magnetic fields, ultimately illustrating how electric currents can be generated through magnetic changes. If you want to delve deeper into its implications or applications, let me know!
The third Maxwell equation is one of the four fundamental equations in Maxwell's set of equations, which describe classical electromagnetism. Specifically, the third Maxwell equation is known as Faraday's Law of Induction. It relates the changing magnetic field to the induced electric field.
In differential form, Faraday's Law of Induction is expressed as:
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
Here’s a breakdown of the components:
- \( \nabla \times \mathbf{E} \) denotes the curl of the electric field \( \mathbf{E} \). This represents the tendency of the electric field to circulate around a point.
- \( \frac{\partial \mathbf{B}}{\partial t} \) is the time rate of change of the magnetic field \( \mathbf{B} \). It indicates how the magnetic field varies with time.
The negative sign in the equation indicates that the induced electric field is oriented in such a way that it opposes the change in the magnetic field (as per Lenz's Law).
In integral form, Faraday's Law is given by:
\[ \oint_{\partial S} \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \int_{S} \mathbf{B} \cdot d\mathbf{A} \]
In this form:
- The left side of the equation, \( \oint_{\partial S} \mathbf{E} \cdot d\mathbf{l} \), represents the electromotive force (EMF) around a closed loop.
- The right side, \( -\frac{d}{dt} \int_{S} \mathbf{B} \cdot d\mathbf{A} \), represents the rate of change of the magnetic flux through a surface \( S \) bounded by the loop.
Faraday's Law is fundamental in understanding how electric generators work, as well as in the operation of transformers and inductors. It also lays the groundwork for the concept of electromagnetic induction.
In differential form, Faraday's Law of Induction is expressed as:
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
Here’s a breakdown of the components:
- \( \nabla \times \mathbf{E} \) denotes the curl of the electric field \( \mathbf{E} \). This represents the tendency of the electric field to circulate around a point.
- \( \frac{\partial \mathbf{B}}{\partial t} \) is the time rate of change of the magnetic field \( \mathbf{B} \). It indicates how the magnetic field varies with time.
The negative sign in the equation indicates that the induced electric field is oriented in such a way that it opposes the change in the magnetic field (as per Lenz's Law).
In integral form, Faraday's Law is given by:
\[ \oint_{\partial S} \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \int_{S} \mathbf{B} \cdot d\mathbf{A} \]
In this form:
- The left side of the equation, \( \oint_{\partial S} \mathbf{E} \cdot d\mathbf{l} \), represents the electromotive force (EMF) around a closed loop.
- The right side, \( -\frac{d}{dt} \int_{S} \mathbf{B} \cdot d\mathbf{A} \), represents the rate of change of the magnetic flux through a surface \( S \) bounded by the loop.
Faraday's Law is fundamental in understanding how electric generators work, as well as in the operation of transformers and inductors. It also lays the groundwork for the concept of electromagnetic induction.