What is Maxwell's equation in free space?
2 Answers
Maxwell's equations describe how electric and magnetic fields interact and propagate. In free space (a vacuum), Maxwell's equations are simplified because there are no charges or currents present. Here are the four Maxwell's equations in free space:
1. **Gauss's Law for Electricity**:
\[
\nabla \cdot \mathbf{E} = 0
\]
This states that the divergence of the electric field \(\mathbf{E}\) is zero in free space, implying there are no free electric charges present.
2. **Gauss's Law for Magnetism**:
\[
\nabla \cdot \mathbf{B} = 0
\]
This states that the divergence of the magnetic field \(\mathbf{B}\) is zero, indicating that there are no magnetic monopoles and that the magnetic field lines are continuous loops.
3. **Faraday's Law of Induction**:
\[
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
\]
This indicates that a time-varying magnetic field induces an electric field. The curl of the electric field \(\mathbf{E}\) is equal to the negative rate of change of the magnetic field \(\mathbf{B}\).
4. **Ampère's Law (with Maxwell's correction)**:
\[
\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\]
This shows that a time-varying electric field generates a magnetic field. The curl of the magnetic field \(\mathbf{B}\) is proportional to the rate of change of the electric field \(\mathbf{E}\), with \(\mu_0\) being the permeability of free space and \(\epsilon_0\) the permittivity of free space.
In these equations:
- \(\mathbf{E}\) represents the electric field.
- \(\mathbf{B}\) represents the magnetic field.
- \(\nabla \cdot\) denotes the divergence operator.
- \(\nabla \times\) denotes the curl operator.
- \(\frac{\partial}{\partial t}\) denotes the partial derivative with respect to time.
- \(\mu_0\) (approximately \(4\pi \times 10^{-7}\) H/m) is the permeability of free space.
- \(\epsilon_0\) (approximately \(8.85 \times 10^{-12}\) F/m) is the permittivity of free space.
These equations are foundational to classical electromagnetism and describe how electric and magnetic fields propagate as electromagnetic waves through free space.
1. **Gauss's Law for Electricity**:
\[
\nabla \cdot \mathbf{E} = 0
\]
This states that the divergence of the electric field \(\mathbf{E}\) is zero in free space, implying there are no free electric charges present.
2. **Gauss's Law for Magnetism**:
\[
\nabla \cdot \mathbf{B} = 0
\]
This states that the divergence of the magnetic field \(\mathbf{B}\) is zero, indicating that there are no magnetic monopoles and that the magnetic field lines are continuous loops.
3. **Faraday's Law of Induction**:
\[
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
\]
This indicates that a time-varying magnetic field induces an electric field. The curl of the electric field \(\mathbf{E}\) is equal to the negative rate of change of the magnetic field \(\mathbf{B}\).
4. **Ampère's Law (with Maxwell's correction)**:
\[
\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\]
This shows that a time-varying electric field generates a magnetic field. The curl of the magnetic field \(\mathbf{B}\) is proportional to the rate of change of the electric field \(\mathbf{E}\), with \(\mu_0\) being the permeability of free space and \(\epsilon_0\) the permittivity of free space.
In these equations:
- \(\mathbf{E}\) represents the electric field.
- \(\mathbf{B}\) represents the magnetic field.
- \(\nabla \cdot\) denotes the divergence operator.
- \(\nabla \times\) denotes the curl operator.
- \(\frac{\partial}{\partial t}\) denotes the partial derivative with respect to time.
- \(\mu_0\) (approximately \(4\pi \times 10^{-7}\) H/m) is the permeability of free space.
- \(\epsilon_0\) (approximately \(8.85 \times 10^{-12}\) F/m) is the permittivity of free space.
These equations are foundational to classical electromagnetism and describe how electric and magnetic fields propagate as electromagnetic waves through free space.
Maxwell's equations in free space (vacuum) describe how electric and magnetic fields propagate. They are:
1. **Gauss's Law for Electricity**: \(\nabla \cdot \mathbf{E} = 0\), indicating that there are no free electric charges in free space.
2. **Gauss's Law for Magnetism**: \(\nabla \cdot \mathbf{B} = 0\), stating that there are no magnetic monopoles.
3. **Faraday's Law of Induction**: \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\), showing that a changing magnetic field creates an electric field.
4. **Ampère's Law (with Maxwell's Addition)**: \(\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\), where \(\mu_0\) and \(\epsilon_0\) are the permeability and permittivity of free space, respectively. This indicates that a changing electric field creates a magnetic field.
These equations are fundamental to understanding electromagnetic waves and fields in a vacuum.
1. **Gauss's Law for Electricity**: \(\nabla \cdot \mathbf{E} = 0\), indicating that there are no free electric charges in free space.
2. **Gauss's Law for Magnetism**: \(\nabla \cdot \mathbf{B} = 0\), stating that there are no magnetic monopoles.
3. **Faraday's Law of Induction**: \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\), showing that a changing magnetic field creates an electric field.
4. **Ampère's Law (with Maxwell's Addition)**: \(\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\), where \(\mu_0\) and \(\epsilon_0\) are the permeability and permittivity of free space, respectively. This indicates that a changing electric field creates a magnetic field.
These equations are fundamental to understanding electromagnetic waves and fields in a vacuum.