1. Scalar and Vector Potentials

In classical field theory (electromagnetism, fluid dynamics, gravitation), many vector fields can be expressed in terms of potentials.

1.1 Scalar Potential $\phi$

A vector field $\vec{F}$ is said to have a scalar potential if it can be written as:

\[\vec{F} = -\nabla \phi\]

Such a field is called:

• Irrotational (curl-free), because
  \(\nabla \times \vec{F} = \nabla \times (-\nabla \phi) = 0\)

Common examples:

• Gravitational field:
  \(\vec{g} = -\nabla \phi_g\)
• Electrostatic field:
  \(\vec{E} = -\nabla V\)

Here, equipotential surfaces are surfaces where $\phi = \text{constant}$ and $\vec{F}$ is always normal to these surfaces.

1.2 Vector Potential $\vec{A}$

A vector field $\vec{F}$ is said to have a vector potential if:

\[\vec{F} = \nabla \times \vec{A}\]

Such fields are:

• Solenoidal (divergence-free), because
  \(\nabla \cdot \vec{F} = \nabla \cdot (\nabla \times \vec{A}) = 0\)

Examples:

• Magnetic field: \(\vec{B} = \nabla \times \vec{A}\)
• Incompressible fluid velocity field: \(\vec{v} = \nabla \times \vec{\Psi}\)

Scalar potentials relate to curl-free fields, vector potentials relate to divergence-free fields.

2. Gauge Freedom

Potentials are not unique.

2.1 Scalar Potential Gauge

If: \(\vec{F} = -\nabla \phi\)

Then adding a constant does not change the field: \(\phi' = \phi + C\)

2.2 Vector Potential Gauge

A new vector potential: \(\vec{A}' = \vec{A} + \nabla \chi\)

still satisfies: \(\nabla \times \vec{A}' = \nabla \times \vec{A}\)

This is the basis of gauge transformations in electromagnetism.

3. Laplacian Operator

The Laplacian is a second-order differential operator defined as:

3.1 Laplacian of a Scalar Field

\(\nabla^2 \phi = \nabla \cdot (\nabla \phi)\)

3.2 Laplacian of a Vector Field

Applied component-wise:

\[\nabla^2 \vec{A} = \left( \nabla^2 A_x \right)\hat{i} + \left( \nabla^2 A_y \right)\hat{j} + \left( \nabla^2 A_z \right)\hat{k}\]

3.3 Physical Meaning

• In electrostatics:
  \(\nabla^2 V = -\frac{\rho}{\epsilon_0}\) (Poisson equation)

• In free space:
  \(\nabla^2 V = 0\) (Laplace equation → harmonic functions)

• In diffusion:
  \(\frac{\partial u}{\partial t} = D \nabla^2 u\)

The Laplacian measures how a quantity “spreads” relative to surrounding values.

Relation Between Potentials and Laplacian

From a scalar potential representation of a field: \(\vec{F} = -\nabla \phi\)

Taking the divergence of both sides gives the relation between the divergence of the field and the Laplacian of the potential: \(\nabla \cdot \vec{F} = -\nabla^2 \phi\)

For a solenoidal field that can be expressed using a vector potential: \(\vec{F} = \nabla \times \vec{A}\)

Taking the curl again yields: \(\nabla \times \vec{F} = \nabla \times (\nabla \times \vec{A})\)

Using the vector identity for the curl of a curl: \(\nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - \nabla^2 \vec{A}\)

This naturally introduces gauge conditions that simplify the equations. Common choices are:

• Coulomb gauge: \(\nabla \cdot \vec{A} = 0\)

This choice eliminates the divergence term completely: \(\nabla \cdot \vec{A} = 0\) Substituting this into the identity simplifies the curl–curl expression to: \(\nabla \times (\nabla \times \vec{A}) = -\nabla^2 \vec{A}\)

• Lorenz gauge: \(\nabla \cdot \vec{A} + \frac{1}{c^2}\frac{\partial \phi}{\partial t} = 0\)

Instead of setting the divergence to zero, this gauge relates it to the scalar potential: \(\nabla \cdot \vec{A} + \frac{1}{c^2}\frac{\partial \phi}{\partial t} = 0\) This choice ensures both $\phi$ and $\vec{A}$ obey wave equations.

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Supplementary

To understand how the Lorenz gauge works, begin with Maxwell’s equations written in terms of potentials:

The fields are: \(\vec{E} = -\nabla \phi - \frac{\partial \vec{A}}{\partial t}\) \(\vec{B} = \nabla \times \vec{A}\)

Insert these potentials into Maxwell’s equations.

From Gauss’s law:

\(\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}\)

Substitute $\vec{E}$:

\[\nabla \cdot \left( -\nabla \phi - \frac{\partial \vec{A}}{\partial t} \right) = \frac{\rho}{\epsilon_0}\]

This gives:

\[-\nabla^2 \phi - \frac{\partial}{\partial t}(\nabla \cdot \vec{A}) = \frac{\rho}{\epsilon_0}\]

Now apply Lorenz gauge:

\[\nabla \cdot \vec{A} = -\frac{1}{c^2}\frac{\partial \phi}{\partial t}\]

Substitute it:

\[-\nabla^2 \phi + \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = \frac{\rho}{\epsilon_0}\]

Rearranging:

\[\nabla^2 \phi - \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = -\frac{\rho}{\epsilon_0}\]

This is a wave equation for the scalar potential.

From Ampère–Maxwell law:

\(\nabla \times \vec{B} - \frac{1}{c^2}\frac{\partial \vec{E}}{\partial t} = \mu_0 \vec{J}\)

Insert potentials:

\[\nabla \times (\nabla \times \vec{A}) + \frac{1}{c^2}\frac{\partial}{\partial t} \left( \nabla \phi + \frac{\partial \vec{A}}{\partial t} \right) = \mu_0 \vec{J}\]

Use vector identity:

\[\nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - \nabla^2 \vec{A}\]

Apply Lorenz gauge again:

\[\nabla(\nabla \cdot \vec{A}) = -\frac{1}{c^2}\nabla \left( \frac{\partial \phi}{\partial t} \right)\]

This cancels with a term from the time derivative of $\vec{E}$.

What remains is:

\[-\nabla^2 \vec{A} + \frac{1}{c^2}\frac{\partial^2 \vec{A}}{\partial t^2} = \mu_0 \vec{J}\]

Or:

\[\nabla^2 \vec{A} - \frac{1}{c^2}\frac{\partial^2 \vec{A}}{\partial t^2} = -\mu_0 \vec{J}\]

This is a wave equation for the vector potential.

Why Lorenz Gauge Works

The Lorenz gauge:

\[\nabla \cdot \vec{A} + \frac{1}{c^2}\frac{\partial \phi}{\partial t} = 0\]

eliminates mixed terms like
$\frac{\partial}{\partial t}(\nabla \cdot \vec{A})$ and
$\nabla (\partial \phi/\partial t)$.

It decouples the potentials and converts Maxwell’s equations into clean wave equations:

• For $ \phi $:

  \[\nabla^2 \phi - \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = -\frac{\rho}{\epsilon_0}\]

• For $ \vec{A} $:

  \[\nabla^2 \vec{A} - \frac{1}{c^2}\frac{\partial^2 \vec{A}}{\partial t^2} = -\mu_0 \vec{J}\]