7.3 Magnetic vector potential

The magnetic vector potential offers a powerful tool for describing magnetic fields. It simplifies calculations and provides insights into electromagnetic phenomena. This concept is crucial for understanding the behavior of magnetic fields in various situations.

By expressing the magnetic field as the curl of the vector potential, we can ensure the field is divergence-free. This approach is particularly useful when dealing with time-varying fields and current distributions in electromagnetism.

Definition of magnetic vector potential

• The magnetic vector potential, denoted as $\vec{A}$, is a vector field that provides an alternative mathematical description of the magnetic field $\vec{B}$
• It simplifies the calculation of the magnetic field in many situations, especially when dealing with time-varying fields or in the presence of currents
• The magnetic vector potential is not uniquely defined, as it is subject to gauge transformations that do not affect the physical magnetic field

Relationship to magnetic field

• The magnetic field $\vec{B}$ can be expressed as the curl of the magnetic vector potential $\vec{A}$: $\vec{B} = \nabla \times \vec{A}$
• This relationship ensures that the magnetic field is divergence-free ($\nabla \cdot \vec{B} = 0$), which is a fundamental property of magnetic fields in the absence of magnetic monopoles
• The choice of the magnetic vector potential is not unique, as different vector potentials can give rise to the same magnetic field through gauge transformations

Gauge transformations

• Gauge transformations are mathematical transformations that change the magnetic vector potential without altering the physical magnetic field
• They are possible because the magnetic field is determined by the curl of the vector potential, which is invariant under the addition of the gradient of a scalar function
• Two common gauge choices are the Coulomb gauge and the Lorenz gauge, each with its own advantages depending on the problem at hand

Coulomb gauge

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• The Coulomb gauge imposes the condition $\nabla \cdot \vec{A} = 0$, which means that the magnetic vector potential is divergence-free
• This gauge is often used in magnetostatics and can simplify the equations by eliminating the scalar potential
• In the Coulomb gauge, the vector potential satisfies Poisson's equation: $\nabla^2 \vec{A} = -\mu_0 \vec{J}$, where $\vec{J}$ is the current density

Lorenz gauge

• The Lorenz gauge is defined by the condition $\nabla \cdot \vec{A} + \frac{1}{c^2} \frac{\partial \phi}{\partial t} = 0$, where $\phi$ is the electric scalar potential and $c$ is the speed of light
• This gauge is particularly useful in electrodynamics, as it leads to symmetric and relativistically covariant equations for the potentials
• In the Lorenz gauge, both the vector and scalar potentials satisfy inhomogeneous wave equations, which describe the propagation of electromagnetic waves

Calculation from current distribution

• The magnetic vector potential can be calculated from the distribution of electric currents using the Biot-Savart law for vector potential
• This law provides a way to determine the vector potential at a point in space due to a given current distribution
• The calculation involves an integral over the current distribution, taking into account the distance between the current elements and the point of interest

Biot-Savart law for vector potential

• The Biot-Savart law for vector potential states that the magnetic vector potential $\vec{A}(\vec{r})$ at a point $\vec{r}$ due to a current distribution $\vec{J}(\vec{r}')$ is given by: $\vec{A}(\vec{r}) = \frac{\mu_0}{4\pi} \int \frac{\vec{J}(\vec{r}')}{|\vec{r} - \vec{r}'|} d^3\vec{r}'$
• Here, $\mu_0$ is the magnetic permeability of free space, and the integral is performed over the volume containing the current distribution
• The Biot-Savart law for vector potential is analogous to the Biot-Savart law for magnetic field, but it calculates the vector potential instead of the field itself

Examples of simple current distributions

• For a straight, infinitely long wire carrying a current $I$, the magnetic vector potential at a distance $r$ from the wire is given by: $\vec{A}(\vec{r}) = \frac{\mu_0 I}{2\pi r} \hat{\phi}$, where $\hat{\phi}$ is the unit vector in the azimuthal direction
• For a circular loop of radius $R$ carrying a current $I$, the magnetic vector potential on the axis of the loop at a distance $z$ from its center is: $\vec{A}(\vec{r}) = \frac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}} \hat{z}$, where $\hat{z}$ is the unit vector along the axis of the loop

Boundary conditions

• The magnetic vector potential must satisfy certain boundary conditions at the interface between different media or regions with different current distributions
• These boundary conditions ensure the continuity of the magnetic field and the conservation of magnetic flux
• The two main boundary conditions for the magnetic vector potential are the continuity of the normal component and the discontinuity of the tangential component

Continuity of normal component

• The normal component of the magnetic vector potential, $\vec{A} \cdot \hat{n}$, must be continuous across the boundary between two different media
• This condition ensures that the normal component of the magnetic field, $\vec{B} \cdot \hat{n}$, is also continuous, as required by the absence of magnetic monopoles
• Mathematically, this boundary condition can be expressed as: $(\vec{A}_1 - \vec{A}_2) \cdot \hat{n} = 0$, where $\vec{A}_1$ and $\vec{A}_2$ are the magnetic vector potentials on either side of the boundary, and $\hat{n}$ is the unit normal vector to the boundary

Discontinuity of tangential component

• The tangential component of the magnetic vector potential, $\vec{A} \times \hat{n}$, can be discontinuous across the boundary between two regions with different current distributions
• The discontinuity is related to the surface current density $\vec{K}$ at the boundary, which is the limiting case of a thin current-carrying layer
• The boundary condition for the tangential component of the magnetic vector potential is given by: $(\vec{A}_1 - \vec{A}_2) \times \hat{n} = \mu_0 \vec{K}$, where $\vec{K}$ is the surface current density at the boundary

Poisson's equation for vector potential

• Poisson's equation is a partial differential equation that relates the magnetic vector potential to the current density distribution
• It can be derived from Maxwell's equations and provides a way to calculate the vector potential from the given current distribution
• Poisson's equation for vector potential is particularly useful in magnetostatics, where the current density is time-independent

Derivation from Maxwell's equations

• Starting from the Maxwell-Ampère law, $\nabla \times \vec{B} = \mu_0 \vec{J}$, and using the definition of the magnetic vector potential, $\vec{B} = \nabla \times \vec{A}$, we can write: $\nabla \times (\nabla \times \vec{A}) = \mu_0 \vec{J}$
• Using the vector identity $\nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - \nabla^2 \vec{A}$ and choosing the Coulomb gauge, $\nabla \cdot \vec{A} = 0$, we arrive at Poisson's equation for vector potential: $\nabla^2 \vec{A} = -\mu_0 \vec{J}$

Solution methods

• Poisson's equation for vector potential can be solved using various methods, depending on the symmetry of the problem and the boundary conditions
• For simple geometries, such as spherical or cylindrical symmetry, the equation can be solved analytically using separation of variables or Green's functions
• For more complex geometries, numerical methods such as finite difference, finite element, or boundary element methods can be employed
• The solution of Poisson's equation yields the magnetic vector potential, from which the magnetic field can be obtained by taking the curl

Magnetic vector potential in magnetostatics

• In magnetostatics, where the current density is time-independent, the magnetic vector potential provides a convenient way to calculate the magnetic field in various configurations
• Two common examples of magnetostatic systems are current-carrying wires and solenoids or toroids, where the magnetic vector potential can be determined analytically or numerically

Current-carrying wires

• For a straight, infinitely long wire carrying a steady current $I$, the magnetic vector potential at a distance $r$ from the wire is given by: $\vec{A}(\vec{r}) = \frac{\mu_0 I}{2\pi r} \hat{\phi}$, where $\hat{\phi}$ is the unit vector in the azimuthal direction
• The magnetic field can be obtained by taking the curl of the vector potential, yielding the well-known result: $\vec{B}(\vec{r}) = \frac{\mu_0 I}{2\pi r} \hat{\phi}$
• For finite-length wires or more complex wire configurations, the magnetic vector potential can be calculated using the Biot-Savart law for vector potential and numerically integrated

Solenoids and toroids

• Solenoids and toroids are examples of magnetostatic systems with a high degree of symmetry, where the magnetic vector potential can be determined analytically
• For an ideal solenoid with $N$ turns, length $L$, and radius $R$, carrying a current $I$, the magnetic vector potential inside the solenoid is given by: $\vec{A}(\vec{r}) = \frac{\mu_0 N I}{L} r \hat{\phi}$, where $r$ is the radial distance from the axis and $\hat{\phi}$ is the azimuthal unit vector
• For a toroid with $N$ turns, major radius $R$, and minor radius $a$, carrying a current $I$, the magnetic vector potential inside the toroid is: $\vec{A}(\vec{r}) = \frac{\mu_0 N I}{2\pi} \ln\left(\frac{R + r}{R - r}\right) \hat{\phi}$, where $r$ is the distance from the toroid's center in the plane of the toroid

Faraday's law in terms of vector potential

• Faraday's law of induction describes the relationship between a time-varying magnetic field and the induced electric field
• In terms of the magnetic vector potential, Faraday's law takes a particularly simple form, highlighting the role of the vector potential in the induction of electric fields
• The formulation of Faraday's law using the magnetic vector potential also demonstrates the gauge invariance of the induced electromotive force (EMF)

Induced electric field

• Faraday's law states that a time-varying magnetic field induces an electric field. In terms of the magnetic vector potential, the induced electric field $\vec{E}$ is given by: $\vec{E} = -\frac{\partial \vec{A}}{\partial t} - \nabla \phi$, where $\phi$ is the electric scalar potential
• The first term, $-\frac{\partial \vec{A}}{\partial t}$, represents the contribution of the time-varying magnetic vector potential to the induced electric field
• The second term, $-\nabla \phi$, is the conservative part of the electric field, which can be eliminated by a gauge transformation of the potentials

Gauge invariance of induced EMF

• The induced electromotive force (EMF) in a closed loop is given by the line integral of the electric field along the loop: $\mathcal{E} = \oint \vec{E} \cdot d\vec{l}$
• Using the expression for the induced electric field in terms of the magnetic vector potential, we can write: $\mathcal{E} = -\oint \frac{\partial \vec{A}}{\partial t} \cdot d\vec{l} - \oint \nabla \phi \cdot d\vec{l}$
• The second term vanishes due to the conservative nature of the scalar potential, leaving: $\mathcal{E} = -\oint \frac{\partial \vec{A}}{\partial t} \cdot d\vec{l}$
• This result demonstrates that the induced EMF depends only on the time variation of the magnetic vector potential and is invariant under gauge transformations of the potentials

Magnetic energy in terms of vector potential

• The magnetic energy stored in a system can be expressed in terms of the magnetic vector potential, providing an alternative formulation to the usual expression in terms of the magnetic field
• This formulation is particularly useful when dealing with current distributions and can simplify the calculation of the magnetic energy in certain cases
• The magnetic energy can be considered in terms of the energy density and the total stored magnetic energy

Energy density

• The magnetic energy density, denoted as $u_m$, is the energy stored in the magnetic field per unit volume
• In terms of the magnetic vector potential and the current density, the magnetic energy density is given by: $u_m = \frac{1}{2} \vec{J} \cdot \vec{A}$
• This expression highlights the role of the current density and the magnetic vector potential in the storage of magnetic energy
• The magnetic energy density can be integrated over the volume containing the current distribution to obtain the total stored magnetic energy

Total stored magnetic energy

• The total stored magnetic energy, denoted as $U_m$, is the energy stored in the magnetic field of a system
• In terms of the magnetic vector potential and the current density, the total stored magnetic energy is given by: $U_m = \frac{1}{2} \int \vec{J} \cdot \vec{A} \, d^3\vec{r}$, where the integral is performed over the volume containing the current distribution
• This expression can be used to calculate the magnetic energy stored in various systems, such as solenoids, toroids, or more complex current distributions
• The total stored magnetic energy is an important quantity in the design and analysis of magnetic systems, as it determines the energy requirements and the potential for energy storage or conversion

Multipole expansion of vector potential

• The multipole expansion is a technique used to approximate the magnetic vector potential of a localized current distribution at large distances from the source
• This expansion is particularly useful when studying the far-field behavior of magnetic systems or when simplifying the calculation of the magnetic field in complex geometries
• The multipole expansion of the magnetic vector potential includes terms such as the dipole, quadrupole, and higher-order terms, each with its own characteristic spatial dependence

Dipole term

• The dipole term is the leading-order term in the multipole expansion of the magnetic vector potential
• It represents the contribution of a magnetic dipole, which is the simplest type of magnetic source
• The dipole term of the magnetic vector potential is given by: $\vec{A}_\text{dipole}(\vec{r}) = \frac{\mu_0}{4\pi} \frac{\vec{m} \times \hat{r}}{r^2}$, where $\vec{m}$ is the magnetic dipole moment, $\hat{r}$ is the unit vector pointing from the dipole to the point of interest, and $r$ is the distance between them
• The magnetic field generated by a magnetic dipole can be obtained by taking the curl of the dipole term of the vector potential

Quadrupole and higher-order terms

• The quadrupole term is the next-order term in the multipole expansion of the magnetic vector potential, representing the contribution of a magnetic quadrupole
• The quadrupole term is given by: $\vec{A}_\text{quadrupole}(\vec{r}) = \frac{\mu_0}{4\pi} \frac{3(\vec{Q} \cdot \hat{r})\hat{r} - \vec{Q}}{r^3}$, where $\vec{Q}$ is the magnetic quadrupole moment tensor
• Higher-order terms, such as octupole and hexadecapole terms, can be included in the multipole expansion for more accurate approximations of the magnetic vector potential
• These higher-order terms become increasingly important when describing the far-field behavior of complex current distributions or when high precision is required

Magnetic vector potential in electrodynamics

• In electrodynamics, where time-varying fields and relativistic effects are considered, the magnetic vector potential plays a crucial role in the description of electromagnetic phenomena
• The equations governing the behavior of the magnetic vector potential in electrodynamics are more complex than in magnetostatics, as they must account for the propagation of electromagnetic waves and the relativistic transformations of fields and potentials
• Two important concepts in this context are the retarded vector potential and Jefimenko's equations

Retarded vector potential

• The retarded vector potential is the magnetic vector potential that takes into account the finite speed of propagation of electromagnetic fields
• It is given by: $\vec{A}(\vec{r}, t) = \frac{\mu_0}{4\pi} \int \frac{\vec{J}(\vec{r}', t - |\vec{r} - \vec{r}'|/c)}{|\vec{r} - \vec{r}'|} d^3\vec{r}'$, where $\