What is the relation between electric field and vector potential?
2 Answers
The relation between the **electric field** (\(\mathbf{E}\)) and the **vector potential** (\(\mathbf{A}\)) is an important concept in electromagnetism, particularly in the context of Maxwell's equations and gauge theory. To understand this relationship, it's necessary to introduce both the scalar and vector potentials used to describe electromagnetic fields.
### 1. **Vector Potential (\(\mathbf{A}\)) and Scalar Potential (\(\phi\))**
In electromagnetism, the electric and magnetic fields can be expressed in terms of the **scalar potential** (\(\phi\)) and the **vector potential** (\(\mathbf{A}\)).
- The **scalar potential** (\(\phi\)) is related to the electrostatic field.
- The **vector potential** (\(\mathbf{A}\)) is related to the magnetic field.
The electric and magnetic fields are then defined as:
- **Magnetic field** (\(\mathbf{B}\)) is given by:
\[
\mathbf{B} = \nabla \times \mathbf{A}
\]
where \(\nabla \times\) is the curl operator.
- **Electric field** (\(\mathbf{E}\)) is given by:
\[
\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}
\]
where \(\frac{\partial \mathbf{A}}{\partial t}\) is the time derivative of the vector potential.
### 2. **Physical Interpretation of the Relation**
The electric field can be thought of as having two components:
- A **static component** due to the **scalar potential** (\(\phi\)).
- A **dynamic component** due to the **time variation of the vector potential** (\(\mathbf{A}\)).
#### Static Field Component:
The term \(-\nabla \phi\) represents the contribution of the scalar potential to the electric field, which corresponds to the electrostatic field in a static charge distribution.
#### Dynamic Field Component:
The term \(-\frac{\partial \mathbf{A}}{\partial t}\) represents the contribution to the electric field due to the time variation of the vector potential. This is particularly important in dynamic situations, such as electromagnetic waves, where time-varying magnetic fields (described by \(\mathbf{A}\)) produce time-varying electric fields.
### 3. **Electromagnetic Waves and Relation Between \(\mathbf{E}\) and \(\mathbf{A}\)**
In dynamic situations, such as the propagation of electromagnetic waves, both the vector potential \(\mathbf{A}\) and scalar potential \(\phi\) vary with time and position. According to Maxwell's equations, a time-varying magnetic field generates an electric field, and this interaction is captured by the term \(-\frac{\partial \mathbf{A}}{\partial t}\).
For example, in free space, the vector potential \(\mathbf{A}\) and scalar potential \(\phi\) can be used to describe the propagation of electromagnetic waves, where the changing \(\mathbf{A}\) generates an electric field and vice versa.
### 4. **Gauge Freedom and Potentials**
There is some freedom in the choice of the scalar and vector potentials due to what is called **gauge invariance**. This means that different choices of \(\phi\) and \(\mathbf{A}\) can yield the same physical electric and magnetic fields. This freedom is used in various gauges, such as the **Coulomb gauge** or the **Lorenz gauge**, to simplify calculations or satisfy specific conditions.
### 5. **Summary of Key Relations**
- The electric field (\(\mathbf{E}\)) is related to both the scalar potential (\(\phi\)) and the vector potential (\(\mathbf{A}\)) as follows:
\[
\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}
\]
- The magnetic field (\(\mathbf{B}\)) is related to the vector potential (\(\mathbf{A}\)):
\[
\mathbf{B} = \nabla \times \mathbf{A}
\]
- The time variation of the vector potential \(\mathbf{A}\) contributes to the generation of an electric field, especially in dynamic situations like electromagnetic waves.
In conclusion, the electric field is influenced by both the scalar potential and the time-varying vector potential, with each contributing to different aspects of the field under static and dynamic conditions.
### 1. **Vector Potential (\(\mathbf{A}\)) and Scalar Potential (\(\phi\))**
In electromagnetism, the electric and magnetic fields can be expressed in terms of the **scalar potential** (\(\phi\)) and the **vector potential** (\(\mathbf{A}\)).
- The **scalar potential** (\(\phi\)) is related to the electrostatic field.
- The **vector potential** (\(\mathbf{A}\)) is related to the magnetic field.
The electric and magnetic fields are then defined as:
- **Magnetic field** (\(\mathbf{B}\)) is given by:
\[
\mathbf{B} = \nabla \times \mathbf{A}
\]
where \(\nabla \times\) is the curl operator.
- **Electric field** (\(\mathbf{E}\)) is given by:
\[
\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}
\]
where \(\frac{\partial \mathbf{A}}{\partial t}\) is the time derivative of the vector potential.
### 2. **Physical Interpretation of the Relation**
The electric field can be thought of as having two components:
- A **static component** due to the **scalar potential** (\(\phi\)).
- A **dynamic component** due to the **time variation of the vector potential** (\(\mathbf{A}\)).
#### Static Field Component:
The term \(-\nabla \phi\) represents the contribution of the scalar potential to the electric field, which corresponds to the electrostatic field in a static charge distribution.
#### Dynamic Field Component:
The term \(-\frac{\partial \mathbf{A}}{\partial t}\) represents the contribution to the electric field due to the time variation of the vector potential. This is particularly important in dynamic situations, such as electromagnetic waves, where time-varying magnetic fields (described by \(\mathbf{A}\)) produce time-varying electric fields.
### 3. **Electromagnetic Waves and Relation Between \(\mathbf{E}\) and \(\mathbf{A}\)**
In dynamic situations, such as the propagation of electromagnetic waves, both the vector potential \(\mathbf{A}\) and scalar potential \(\phi\) vary with time and position. According to Maxwell's equations, a time-varying magnetic field generates an electric field, and this interaction is captured by the term \(-\frac{\partial \mathbf{A}}{\partial t}\).
For example, in free space, the vector potential \(\mathbf{A}\) and scalar potential \(\phi\) can be used to describe the propagation of electromagnetic waves, where the changing \(\mathbf{A}\) generates an electric field and vice versa.
### 4. **Gauge Freedom and Potentials**
There is some freedom in the choice of the scalar and vector potentials due to what is called **gauge invariance**. This means that different choices of \(\phi\) and \(\mathbf{A}\) can yield the same physical electric and magnetic fields. This freedom is used in various gauges, such as the **Coulomb gauge** or the **Lorenz gauge**, to simplify calculations or satisfy specific conditions.
### 5. **Summary of Key Relations**
- The electric field (\(\mathbf{E}\)) is related to both the scalar potential (\(\phi\)) and the vector potential (\(\mathbf{A}\)) as follows:
\[
\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}
\]
- The magnetic field (\(\mathbf{B}\)) is related to the vector potential (\(\mathbf{A}\)):
\[
\mathbf{B} = \nabla \times \mathbf{A}
\]
- The time variation of the vector potential \(\mathbf{A}\) contributes to the generation of an electric field, especially in dynamic situations like electromagnetic waves.
In conclusion, the electric field is influenced by both the scalar potential and the time-varying vector potential, with each contributing to different aspects of the field under static and dynamic conditions.
The electric field \(\mathbf{E}\) and the vector potential \(\mathbf{A}\) are related through Maxwell's equations, specifically in the context of electromagnetism. To understand their relationship, it’s essential to look at how these quantities are defined and how they interact in different scenarios. Here’s a detailed explanation:
### Electric Field and Vector Potential
1. **Definitions:**
- **Electric Field (\(\mathbf{E}\)):** The electric field represents the force per unit charge exerted on a stationary charge in space. It is defined as the negative gradient of the electric potential \(V\), plus the partial derivative of the vector potential \(\mathbf{A}\) with respect to time:
\[
\mathbf{E} = -\nabla V - \frac{\partial \mathbf{A}}{\partial t}
\]
- **Vector Potential (\(\mathbf{A}\)):** The vector potential is a vector field whose curl gives the magnetic field \(\mathbf{B}\). It is related to the magnetic field by:
\[
\mathbf{B} = \nabla \times \mathbf{A}
\]
2. **Electromagnetic Fields and Potentials:**
- Maxwell's equations describe how electric and magnetic fields propagate and interact with charges and currents. These fields can be derived from scalar and vector potentials. For time-dependent fields, the potentials \(\Phi\) (scalar potential) and \(\mathbf{A}\) (vector potential) are used. The electric and magnetic fields are then given by:
\[
\mathbf{E} = -\nabla \Phi - \frac{\partial \mathbf{A}}{\partial t}
\]
\[
\mathbf{B} = \nabla \times \mathbf{A}
\]
- Here, \(\Phi\) is related to the scalar potential, and \(\mathbf{A}\) is the vector potential. The term \(- \frac{\partial \mathbf{A}}{\partial t}\) represents the time-varying part of the electric field, which is particularly important in dynamic situations where both electric and magnetic fields change over time.
3. **Relation through Maxwell's Equations:**
- **Faraday’s Law of Induction:** This law states that a changing magnetic field induces an electric field. Mathematically:
\[
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
\]
Using \(\mathbf{B} = \nabla \times \mathbf{A}\), this can be rewritten as:
\[
\nabla \times \mathbf{E} = -\frac{\partial (\nabla \times \mathbf{A})}{\partial t}
\]
Applying vector calculus identities, this relation shows how the electric field is influenced by the time variation of the magnetic field (vector potential).
- **Ampère’s Law (with Maxwell’s correction):** This law relates the magnetic field to the electric current and changing electric fields:
\[
\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\]
Substituting \(\mathbf{B} = \nabla \times \mathbf{A}\) into this equation, we get:
\[
\nabla \times (\nabla \times \mathbf{A}) = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\]
Using vector identities, this shows the relationship between the vector potential \(\mathbf{A}\), current density \(\mathbf{J}\), and the electric field.
4. **Gauge Freedom:**
- The vector potential \(\mathbf{A}\) and the scalar potential \(\Phi\) are not uniquely defined; they can be altered by a gauge transformation without changing the physical fields. For instance, a gauge transformation modifies \(\mathbf{A}\) and \(\Phi\) as follows:
\[
\mathbf{A} \to \mathbf{A} + \nabla \Lambda
\]
\[
\Phi \to \Phi - \frac{\partial \Lambda}{\partial t}
\]
where \(\Lambda\) is any differentiable function. This gauge freedom shows that while \(\mathbf{A}\) and \(\Phi\) can be chosen in various ways, the resulting \(\mathbf{E}\) and \(\mathbf{B}\) fields are invariant.
### Summary
In summary, the electric field \(\mathbf{E}\) and the vector potential \(\mathbf{A}\) are related through Maxwell's equations. The vector potential helps describe the magnetic field \(\mathbf{B}\) and is directly involved in calculating the time-varying component of the electric field. The full electric field is a combination of the gradient of the scalar potential and the time derivative of the vector potential. These relationships are crucial for understanding electromagnetic waves and other dynamic electromagnetic phenomena.
### Electric Field and Vector Potential
1. **Definitions:**
- **Electric Field (\(\mathbf{E}\)):** The electric field represents the force per unit charge exerted on a stationary charge in space. It is defined as the negative gradient of the electric potential \(V\), plus the partial derivative of the vector potential \(\mathbf{A}\) with respect to time:
\[
\mathbf{E} = -\nabla V - \frac{\partial \mathbf{A}}{\partial t}
\]
- **Vector Potential (\(\mathbf{A}\)):** The vector potential is a vector field whose curl gives the magnetic field \(\mathbf{B}\). It is related to the magnetic field by:
\[
\mathbf{B} = \nabla \times \mathbf{A}
\]
2. **Electromagnetic Fields and Potentials:**
- Maxwell's equations describe how electric and magnetic fields propagate and interact with charges and currents. These fields can be derived from scalar and vector potentials. For time-dependent fields, the potentials \(\Phi\) (scalar potential) and \(\mathbf{A}\) (vector potential) are used. The electric and magnetic fields are then given by:
\[
\mathbf{E} = -\nabla \Phi - \frac{\partial \mathbf{A}}{\partial t}
\]
\[
\mathbf{B} = \nabla \times \mathbf{A}
\]
- Here, \(\Phi\) is related to the scalar potential, and \(\mathbf{A}\) is the vector potential. The term \(- \frac{\partial \mathbf{A}}{\partial t}\) represents the time-varying part of the electric field, which is particularly important in dynamic situations where both electric and magnetic fields change over time.
3. **Relation through Maxwell's Equations:**
- **Faraday’s Law of Induction:** This law states that a changing magnetic field induces an electric field. Mathematically:
\[
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
\]
Using \(\mathbf{B} = \nabla \times \mathbf{A}\), this can be rewritten as:
\[
\nabla \times \mathbf{E} = -\frac{\partial (\nabla \times \mathbf{A})}{\partial t}
\]
Applying vector calculus identities, this relation shows how the electric field is influenced by the time variation of the magnetic field (vector potential).
- **Ampère’s Law (with Maxwell’s correction):** This law relates the magnetic field to the electric current and changing electric fields:
\[
\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\]
Substituting \(\mathbf{B} = \nabla \times \mathbf{A}\) into this equation, we get:
\[
\nabla \times (\nabla \times \mathbf{A}) = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\]
Using vector identities, this shows the relationship between the vector potential \(\mathbf{A}\), current density \(\mathbf{J}\), and the electric field.
4. **Gauge Freedom:**
- The vector potential \(\mathbf{A}\) and the scalar potential \(\Phi\) are not uniquely defined; they can be altered by a gauge transformation without changing the physical fields. For instance, a gauge transformation modifies \(\mathbf{A}\) and \(\Phi\) as follows:
\[
\mathbf{A} \to \mathbf{A} + \nabla \Lambda
\]
\[
\Phi \to \Phi - \frac{\partial \Lambda}{\partial t}
\]
where \(\Lambda\) is any differentiable function. This gauge freedom shows that while \(\mathbf{A}\) and \(\Phi\) can be chosen in various ways, the resulting \(\mathbf{E}\) and \(\mathbf{B}\) fields are invariant.
### Summary
In summary, the electric field \(\mathbf{E}\) and the vector potential \(\mathbf{A}\) are related through Maxwell's equations. The vector potential helps describe the magnetic field \(\mathbf{B}\) and is directly involved in calculating the time-varying component of the electric field. The full electric field is a combination of the gradient of the scalar potential and the time derivative of the vector potential. These relationships are crucial for understanding electromagnetic waves and other dynamic electromagnetic phenomena.