What is h in Maxwell's 4th equation?
2 Answers
In Maxwell's equations, the fourth equation is typically expressed in its integral form as:
\[ \oint_{\partial V} \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \int_V \mathbf{B} \cdot d\mathbf{A} \]
This is Faraday's Law of Induction. Here, \( h \) is not a variable in this equation.
However, if you're referring to Maxwell's equations in differential form, the fourth equation is:
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
In this context, \( h \) is still not a part of the equation. The variable \( h \) might be a placeholder or a specific constant in some other context, but it’s not a standard component in Maxwell's four fundamental equations.
\[ \oint_{\partial V} \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \int_V \mathbf{B} \cdot d\mathbf{A} \]
This is Faraday's Law of Induction. Here, \( h \) is not a variable in this equation.
However, if you're referring to Maxwell's equations in differential form, the fourth equation is:
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
In this context, \( h \) is still not a part of the equation. The variable \( h \) might be a placeholder or a specific constant in some other context, but it’s not a standard component in Maxwell's four fundamental equations.
In Maxwell's 4th equation, which is often referred to as **Ampère’s Law with Maxwell's correction**, the symbol **H** represents the **magnetic field intensity** or **magnetic field strength**. It is related to the magnetic effects caused by currents and time-varying electric fields.
### Maxwell's 4th Equation (Ampère’s Law with Maxwell's Correction)
Ampère’s Law with Maxwell’s correction is written as:
\[
\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}
\]
Where:
- \( \nabla \times \mathbf{H} \) is the **curl of the magnetic field intensity** (H).
- \( \mathbf{J} \) is the **current density** (the flow of electric current per unit area).
- \( \frac{\partial \mathbf{D}}{\partial t} \) is the **displacement current density**, which accounts for the time-varying electric field.
### Understanding **H**:
- **H** (magnetic field intensity) describes the response of the material (or medium) to an applied magnetic field. It is related to the strength of the magnetic field in free space and inside materials, but it is independent of the material's magnetic properties.
- **H** is related to another vector field, **B** (magnetic flux density or magnetic induction), through the material's properties, specifically its permeability \( \mu \):
\[
\mathbf{B} = \mu \mathbf{H}
\]
Where:
- **B** is the magnetic flux density (also called the magnetic field).
- \( \mu \) is the permeability of the material (or free space in the simplest case).
In free space, \( \mu = \mu_0 \), where \( \mu_0 \) is the permeability of free space (approximately \( 4\pi \times 10^{-7} \, \text{H/m} \)).
### Role of **H** in Maxwell’s 4th Equation:
- **H** represents how the magnetic field varies in response to electric currents (\( \mathbf{J} \)) and changing electric fields (\( \frac{\partial \mathbf{D}}{\partial t} \)).
- When you have a time-varying electric field, such as in electromagnetic waves, the term \( \frac{\partial \mathbf{D}}{\partial t} \) (displacement current) becomes significant, and this is Maxwell's correction to Ampère’s Law.
Thus, **H** is crucial for describing how magnetic fields interact with electric currents and varying electric fields, playing a central role in the propagation of electromagnetic waves.
### Maxwell's 4th Equation (Ampère’s Law with Maxwell's Correction)
Ampère’s Law with Maxwell’s correction is written as:
\[
\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}
\]
Where:
- \( \nabla \times \mathbf{H} \) is the **curl of the magnetic field intensity** (H).
- \( \mathbf{J} \) is the **current density** (the flow of electric current per unit area).
- \( \frac{\partial \mathbf{D}}{\partial t} \) is the **displacement current density**, which accounts for the time-varying electric field.
### Understanding **H**:
- **H** (magnetic field intensity) describes the response of the material (or medium) to an applied magnetic field. It is related to the strength of the magnetic field in free space and inside materials, but it is independent of the material's magnetic properties.
- **H** is related to another vector field, **B** (magnetic flux density or magnetic induction), through the material's properties, specifically its permeability \( \mu \):
\[
\mathbf{B} = \mu \mathbf{H}
\]
Where:
- **B** is the magnetic flux density (also called the magnetic field).
- \( \mu \) is the permeability of the material (or free space in the simplest case).
In free space, \( \mu = \mu_0 \), where \( \mu_0 \) is the permeability of free space (approximately \( 4\pi \times 10^{-7} \, \text{H/m} \)).
### Role of **H** in Maxwell’s 4th Equation:
- **H** represents how the magnetic field varies in response to electric currents (\( \mathbf{J} \)) and changing electric fields (\( \frac{\partial \mathbf{D}}{\partial t} \)).
- When you have a time-varying electric field, such as in electromagnetic waves, the term \( \frac{\partial \mathbf{D}}{\partial t} \) (displacement current) becomes significant, and this is Maxwell's correction to Ampère’s Law.
Thus, **H** is crucial for describing how magnetic fields interact with electric currents and varying electric fields, playing a central role in the propagation of electromagnetic waves.