What is the 4th Maxwell equation in differential form?
2 Answers
The fourth Maxwell equation in differential form is known as **Faraday's law of induction**, which describes how a changing magnetic field induces an electric field. It is expressed mathematically as:
\[
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
\]
where:
- \(\nabla \times \mathbf{E}\) represents the curl of the electric field \(\mathbf{E}\),
- \(\mathbf{B}\) is the magnetic field,
- \(\frac{\partial \mathbf{B}}{\partial t}\) is the partial derivative of the magnetic field with respect to time.
This equation emphasizes the relationship between electric fields and changing magnetic fields, a fundamental principle in electromagnetism.
\[
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
\]
where:
- \(\nabla \times \mathbf{E}\) represents the curl of the electric field \(\mathbf{E}\),
- \(\mathbf{B}\) is the magnetic field,
- \(\frac{\partial \mathbf{B}}{\partial t}\) is the partial derivative of the magnetic field with respect to time.
This equation emphasizes the relationship between electric fields and changing magnetic fields, a fundamental principle in electromagnetism.
The fourth Maxwell equation in differential form is one of the key equations in Maxwell's set of equations, which describe the fundamental interactions of electric and magnetic fields. The fourth Maxwell equation is specifically known as **Faraday's Law of Induction**.
In differential form, Faraday's Law is expressed as:
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
Here’s a breakdown of this equation:
- \(\nabla \times \mathbf{E}\) represents the curl of the electric field \(\mathbf{E}\). The curl operation measures the tendency of the field to circulate around a point.
- \(\frac{\partial \mathbf{B}}{\partial t}\) is the partial derivative of the magnetic field \(\mathbf{B}\) with respect to time. This term shows how the magnetic field changes over time.
**Interpretation:**
Faraday’s Law of Induction tells us that a time-varying magnetic field induces an electric field. This is a fundamental principle in electromagnetism and is the basis for many practical applications, such as electric generators and transformers.
In other words, a changing magnetic field produces a rotating electric field, and this induced electric field can drive currents in a conductor, illustrating the interplay between electricity and magnetism.
In differential form, Faraday's Law is expressed as:
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
Here’s a breakdown of this equation:
- \(\nabla \times \mathbf{E}\) represents the curl of the electric field \(\mathbf{E}\). The curl operation measures the tendency of the field to circulate around a point.
- \(\frac{\partial \mathbf{B}}{\partial t}\) is the partial derivative of the magnetic field \(\mathbf{B}\) with respect to time. This term shows how the magnetic field changes over time.
**Interpretation:**
Faraday’s Law of Induction tells us that a time-varying magnetic field induces an electric field. This is a fundamental principle in electromagnetism and is the basis for many practical applications, such as electric generators and transformers.
In other words, a changing magnetic field produces a rotating electric field, and this induced electric field can drive currents in a conductor, illustrating the interplay between electricity and magnetism.