7.2 Magnetic scalar potential

Definition of magnetic scalar potential

Magnetic scalar potential is a scalar field, denoted $\Psi$, that lets you describe the magnetic field intensity $\vec{H}$ through a simple gradient relationship in regions free of free currents and time-varying electric fields. It plays the same role for magnetostatics that electric potential $V$ plays in electrostatics: it reduces a vector problem to a scalar one, which is far easier to solve.

This tool shows up constantly when analyzing permanent magnets, magnetic circuits, and shielding problems. Its main limitation is that it only works where $\nabla \times \vec{H} = 0$. Wherever free currents flow, you need the magnetic vector potential $\vec{A}$ instead.

$\Psi$ is measured in amperes (SI) or sometimes quoted in ampere-turns. In CGS, the unit is the gilbert.

Analogy with electric potential

The parallel with electrostatics is direct and worth internalizing:

• In electrostatics, $\vec{E} = -\nabla V$ holds wherever there are no time-varying magnetic fields.
• In magnetostatics, $\vec{H} = -\nabla \Psi$ holds wherever there are no free currents.

Both relationships work because the relevant field is curl-free in its domain of validity. The negative sign means the field points from high potential toward low potential, just as in the electric case.

Mathematical representation

• $\Psi$ is a function of position: $\Psi(\vec{r})$
• The defining relation is $\vec{H} = -\nabla \Psi$
• In a source-free region (no free currents, no magnetization), $\Psi$ satisfies Laplace's equation: $\nabla^2 \Psi = 0$
• In a region containing magnetized material, you can introduce an equivalent magnetic charge density $\rho_m = -\nabla \cdot \vec{M}$, and $\Psi$ then satisfies Poisson's equation: $\nabla^2 \Psi = \nabla \cdot \vec{M}$

Note the sign conventions carefully. The "magnetic charge" picture is a mathematical convenience; there are no real magnetic monopoles. The quantity $\rho_m = -\nabla \cdot \vec{M}$ is sometimes called the volume magnetic pole density.

Laplace's equation for magnetic scalar potential

Laplace's equation $\nabla^2 \Psi = 0$ governs the scalar potential in any region that is both current-free and free of magnetized material (or where $\vec{M}$ is uniform so that $\nabla \cdot \vec{M} = 0$). Solving it is the central task in most scalar-potential problems.

Derivation from Maxwell's equations

1. Start with the magnetostatic curl equation in a current-free region: $\nabla \times \vec{H} = 0$.
2. Because $\vec{H}$ is curl-free, you can write $\vec{H} = -\nabla \Psi$.
3. Now use Gauss's law for magnetism, $\nabla \cdot \vec{B} = 0$. In free space or a linear medium with constant permeability $\mu$, $\vec{B} = \mu \vec{H}$, so $\nabla \cdot \vec{H} = 0$.
4. Substituting the gradient expression: $\nabla \cdot (-\nabla \Psi) = 0$, which gives $\nabla^2 \Psi = 0$.

The key prerequisite is step 1: you must be in a region with no free current density $\vec{J}_f$.

Boundary conditions

Laplace's equation is an elliptic PDE, so you need boundary conditions on the entire boundary of your domain:

• Dirichlet condition: the value of $\Psi$ is specified on the boundary surface.
• Neumann condition: the normal derivative $\partial \Psi / \partial n$ is specified on the boundary surface. Since $H_n = -\partial \Psi / \partial n$, this is equivalent to specifying the normal component of $\vec{H}$.
• Interface condition: at a boundary between two media with permeabilities $\mu_1$ and $\mu_2$, continuity of the normal component of $\vec{B}$ requires $\mu_1 \frac{\partial \Psi_1}{\partial n} = \mu_2 \frac{\partial \Psi_2}{\partial n}$. The tangential component of $\vec{H}$ is continuous (when no surface current is present), so $\Psi$ itself is continuous across the interface.

Poisson's equation for magnetic scalar potential

When magnetized material is present, the divergence of $\vec{M}$ acts as an effective source term, and Laplace's equation generalizes to Poisson's equation.

Derivation from Maxwell's equations

1. In a current-free region, $\vec{H} = -\nabla \Psi$ still holds.
2. Write $\vec{B} = \mu_0(\vec{H} + \vec{M})$ and apply $\nabla \cdot \vec{B} = 0$: $\nabla \cdot \vec{H} = -\nabla \cdot \vec{M}$
3. Substitute $\vec{H} = -\nabla \Psi$: $\nabla^2 \Psi = \nabla \cdot \vec{M}$

Some texts define $\rho_m = -\nabla \cdot \vec{M}$ so the equation looks like the electrostatic Poisson equation: $\nabla^2 \Psi = -\rho_m$. Watch the sign convention in whatever source you're using.

Boundary conditions

The same Dirichlet and Neumann conditions apply as for Laplace's equation. At an interface where the magnetization jumps discontinuously, there is an effective surface magnetic charge density $\sigma_m = (\vec{M}_1 - \vec{M}_2) \cdot \hat{n}$. This produces a jump in the normal derivative of $\Psi$ across the surface:

$\frac{\partial \Psi_2}{\partial n} - \frac{\partial \Psi_1}{\partial n} = \sigma_m$

This is directly analogous to the surface charge jump condition for electric potential.

Presence of magnetic sources

The "magnetic charges" $\rho_m$ and $\sigma_m$ are fictitious. No magnetic monopoles exist. They are a bookkeeping device: wherever the magnetization has a nonzero divergence (volume poles) or a discontinuity at a surface (surface poles), you get an effective source for $\Psi$. This picture is especially useful for permanent magnets, where $\vec{M}$ is given and you want to find the resulting $\vec{H}$ field.

Solution methods for Laplace's and Poisson's equations

The mathematical structure of these equations is identical to what you encounter in electrostatics, so the same toolkit applies.

Separation of variables

This is the go-to analytical method when the geometry matches a standard coordinate system:

1. Choose coordinates that align with the boundary surfaces (Cartesian, cylindrical, or spherical).
2. Assume a product solution, e.g., $\Psi(r,\theta) = R(r)\Theta(\theta)$ in spherical coordinates.
3. Substitute into Laplace's equation and separate into ODEs for each coordinate.
4. Solve each ODE and apply boundary conditions to fix the constants.

Typical applications include the field of a uniformly magnetized sphere (expand in Legendre polynomials), fields inside and outside cylindrical shells, and coaxial geometries.

Green's functions

For Poisson's equation with a distributed source $\rho_m(\vec{r}')$, the solution can be written as an integral:

$\Psi(\vec{r}) = \frac{1}{4\pi} \int \frac{\rho_m(\vec{r}')}{|\vec{r} - \vec{r}'|} \, d^3r'$

This is the magnetic analog of the Coulomb integral for electric potential. Surface pole contributions add a surface integral with $\sigma_m$. The Green's function approach is powerful but requires you to know (or construct) the Green's function for your boundary geometry.

Numerical techniques

When analytical methods fail due to complex geometry or inhomogeneous materials:

• Finite difference method (FDM): discretize the domain on a grid and replace derivatives with difference quotients. Straightforward to implement but less flexible with irregular boundaries.
• Finite element method (FEM): divide the domain into small elements (triangles, tetrahedra) and approximate $\Psi$ with basis functions on each element. Handles complex shapes and material interfaces well. This is what most commercial EM solvers use.
• Boundary element method (BEM): only the boundary surfaces are meshed, reducing a 3D problem to a 2D one. Efficient when the source region is small compared to the total domain.

Magnetic fields from scalar potential

Once you have $\Psi$, extracting the field is straightforward: take the gradient and flip the sign.

Gradient of scalar potential

$\vec{H} = -\nabla \Psi$

In the three standard coordinate systems:

• Cartesian: $\vec{H} = -\left(\frac{\partial \Psi}{\partial x}\hat{x} + \frac{\partial \Psi}{\partial y}\hat{y} + \frac{\partial \Psi}{\partial z}\hat{z}\right)$
• Cylindrical: $\vec{H} = -\left(\frac{\partial \Psi}{\partial s}\hat{s} + \frac{1}{s}\frac{\partial \Psi}{\partial \phi}\hat{\phi} + \frac{\partial \Psi}{\partial z}\hat{z}\right)$
• Spherical: $\vec{H} = -\left(\frac{\partial \Psi}{\partial r}\hat{r} + \frac{1}{r}\frac{\partial \Psi}{\partial \theta}\hat{\theta} + \frac{1}{r\sin\theta}\frac{\partial \Psi}{\partial \phi}\hat{\phi}\right)$

The negative sign means $\vec{H}$ points from regions of high $\Psi$ toward regions of low $\Psi$.

Curl-free nature of magnetic fields

Any field derived from a scalar potential is automatically irrotational:

$\nabla \times \vec{H} = \nabla \times (-\nabla \Psi) = 0$

This is a vector identity, true for any smooth scalar field. It confirms internal consistency: you started by assuming $\nabla \times \vec{H} = 0$ (no free currents), and the scalar potential formulation preserves that condition exactly.

Comparison with vector potential approach

Use $\Psi$ when you can (current-free regions). Use $\vec{A}$ when you must (current sources present, or when you need $\vec{B}$ to be automatically divergence-free).

Applications of magnetic scalar potential

Magnetostatics problems

• Permanent magnets: given a known magnetization $\vec{M}$, compute the equivalent magnetic charges and solve Poisson's equation for $\Psi$. The classic example is the uniformly magnetized sphere, where the internal field turns out to be uniform: $\vec{H}_{in} = -\frac{1}{3}\vec{M}$.
• Magnetic circuits: in high-permeability cores, $\Psi$ drops primarily across air gaps, analogous to voltage drops across resistors. This makes magnetic circuit analysis very natural in the scalar potential picture.
• Magnetic lenses: the focusing fields in electron optics are often computed via $\Psi$ in the current-free bore region.

Shielding and field confinement

High-permeability materials (like mu-metal, with $\mu_r \sim 10^4 - 10^5$) create a low-reluctance path that diverts magnetic flux away from the shielded region. In the scalar potential picture, the potential is nearly constant inside the shield, so $\vec{H} = -\nabla \Psi \approx 0$ there. Designing effective shields amounts to solving a boundary value problem for $\Psi$ with the appropriate permeability contrast.

Magnetic materials and permeability

For linear materials with $\vec{B} = \mu \vec{H}$, the scalar potential formulation remains clean. For nonlinear materials (where $\mu$ depends on $|\vec{H}|$), the equation becomes nonlinear and typically requires iterative numerical solution. Inhomogeneous materials with spatially varying $\mu(\vec{r})$ lead to a modified equation:

$\nabla \cdot [\mu(\vec{r}) \nabla \Psi] = 0$

This replaces the simple Laplacian and must generally be solved numerically.

Limitations of magnetic scalar potential

Inability to handle current sources

The entire formulation rests on $\nabla \times \vec{H} = 0$. The moment free current density $\vec{J}_f$ is present, Ampère's law gives $\nabla \times \vec{H} = \vec{J}_f \neq 0$, and no scalar potential can reproduce a field with nonzero curl. You cannot use $\Psi$ inside or near conductors carrying current.

A partial workaround exists: in problems with localized currents (like a coil), you can define $\Psi$ in the current-free region outside the coil, but you must handle the multivaluedness that arises when the region is not simply connected. This leads to the concept of a reduced scalar potential or total scalar potential with branch cuts, which adds significant complexity.

Need for vector potential in certain cases

When currents or time-varying fields are present, you must use the magnetic vector potential $\vec{A}$, defined by $\vec{B} = \nabla \times \vec{A}$. The vector potential has no restriction on curl and automatically satisfies $\nabla \cdot \vec{B} = 0$. The tradeoff is that you're solving for three component functions instead of one scalar.

Connection with magnetic vector potential

Gauge transformations

The vector potential is not unique. You can replace $\vec{A}$ with $\vec{A}' = \vec{A} + \nabla \psi$ (where $\psi$ is any smooth scalar function) without changing $\vec{B}$, because $\nabla \times (\nabla \psi) = 0$. This freedom is called gauge invariance.

Common gauge choices:

• Coulomb gauge: $\nabla \cdot \vec{A} = 0$. Simplifies magnetostatics; the vector potential satisfies a vector Poisson equation $\nabla^2 \vec{A} = -\mu_0 \vec{J}$.
• Lorenz gauge: $\nabla \cdot \vec{A} = -\mu_0 \varepsilon_0 \frac{\partial \phi}{\partial t}$. Puts the wave equations for $\vec{A}$ and $\phi$ into symmetric form. Relevant for electrodynamics, not magnetostatics.

The scalar potential $\Psi$ has no gauge freedom. It is uniquely determined (up to an additive constant) once boundary conditions are specified.

Helmholtz decomposition theorem

The Helmholtz theorem provides the theoretical foundation connecting the two potential approaches. Any well-behaved vector field $\vec{F}$ that vanishes sufficiently fast at infinity can be decomposed as:

$\vec{F} = -\nabla \Psi + \nabla \times \vec{A}$

• The first term is irrotational (curl-free): this is the part captured by the scalar potential.
• The second term is solenoidal (divergence-free): this is the part captured by the vector potential.

For the magnetic field $\vec{H}$, the irrotational part comes from magnetic "charges" (bound poles in magnetized matter), while the solenoidal part comes from free currents. In a current-free region, only the scalar potential piece survives, which is exactly why $\Psi$ works there.