If the curl of a vector field F vanishes everywhere, then F is known as an irrotational vector field, and it can be written as the gradient of a scalar potential V:
F=−∇V.
If the divergence of a vector field F vanishes everywhere, then F is known as a solenoidal vector field and can be written as the curl of a vector potential A:
F=∇×A.
In fact, any vector field F can be written as a sum of both:
The potential formulation of electromagnetism is very powerful, but it comes with an important subtlety: the potentials are not unique.
Just like many different functions can have the same derivative (differing by a constant), many different pairs of potentials (V,A) can produce the exact same physical electric and magnetic fields (E,B). This redundancy is known as gauge invariance or gauge freedom.
Putting it all together, the gauge transformation laws are:
{A′=A+∇λV′=V−∂t∂λ
where λ(r,t) is an arbitrary scalar gauge function.
This means:
We are free to add the gradient of any scalar field λ to the vector potential A, as long as we simultaneously subtract the time derivative of λ from the scalar potential V.
We’ve seen that the scalar and vector potentials, V and A, are not uniquely defined — there’s a degree of freedom called gauge freedom. That means we are free to perform a gauge transformation using any scalar function λ(r,t):
A′V′=A+∇λ=V−∂t∂λ
This transformation leaves the physical fields E and B unchanged. Since the potentials aren’t unique, we might as well choose them in a way that simplifies the equations. That’s the idea behind fixing a gauge.
One such choice is called the Coulomb gauge (also known as the radiation gauge). In this gauge, we impose the condition:
∇⋅A=0
That is, the divergence of the vector potential vanishes.
Recall the potential formulation of the electric field:
E=−∇V−∂t∂A
Plug this into Gauss’s law:
∇⋅(−∇V−∂t∂A)=ϵ0ρ
Using the divergence of a gradient and the gauge condition:
−∇2V−∂t∂(∇⋅A)=ϵ0ρ⇒−∇2V=ϵ0ρ
So we get:
∇2V=−ϵ0ρ
This is Poisson’s equation, just like in electrostatics. Even though the system may be time-dependent, the scalar potential V is determined instantaneously by the charge distribution. That is, V(r,t) depends on ρ(r,t) at the same time t, without any retardation.
Once we've accepted that the potentials (V,A) are not unique — due to gauge freedom — a natural question arises: can we choose a specific form of the potentials that simplifies our equations?
Yes — and that’s the purpose of choosing a gauge. Let’s now see how to apply this idea to simplify Maxwell’s equations by choosing an appropriate gauge field λ(r,t).
We’ll begin by substituting the potential expressions into the Ampère-Maxwell law.
Recall that:
B=∇×A,E=−∇V−∂t∂A
Plug these into the Ampère-Maxwell law:
∇×B=μ0J+μ0ϵ0∂t∂E
The left-hand side becomes:
∇×(∇×A)=μ0J−μ0ϵ0(∂t∂∇V+∂t2∂2A)
Now use the vector identity:
∇×(∇×A)=∇(∇⋅A)−∇2A
So the equation becomes:
∇(∇⋅A)−∇2A=μ0J−μ0ϵ0(∇∂t∂V+∂t2∂2A)
Let’s bring all terms to one side:
(∇2A−μ0ϵ0∂t2∂2A)−∇(∇⋅A+μ0ϵ0∂t∂V)=−μ0J
At this point, we see a mixed term involving both V and A. To simplify the equations, we choose a gauge that eliminates the coupling term:
∇⋅A=−μ0ϵ0∂t∂V
This condition is called the Lorenz gauge. It ensures that the potentials decouple nicely in Maxwell’s equations and have symmetric wave-equation-like forms.
Under the Lorenz gauge condition, the complicated gradient term in Ampère’s law vanishes, leaving us with:
∇2A−μ0ϵ0∂t2∂2A=−μ0J
We now repeat the procedure for Gauss’s law, starting with:
∇⋅E=ϵ0ρ
Substitute E=−∇V−∂t∂A:
−∇2V−∂t∂(∇⋅A)=ϵ0ρ
Apply the Lorenz gauge condition:
−∇2V+μ0ϵ0∂t2∂2V=ϵ0ρ
This simplifies to:
∇2V−μ0ϵ0∂t2∂2V=−ϵ0ρ
Thus, by choosing the Lorenz gauge we have decoupled Maxwell’s equations into two inhomogeneous wave equations that together contain the entire theory of electromagnetism:
□2V□2A=−ϵ0ρ=−μ0J
where the d’Alembertian operator is defined as:
□2=∇2−μ0ϵ0∂t2∂2
This formulation not only simplifies the math but also exposes the deep symmetry between the scalar and vector potentials in electromagnetism. It also sets the stage for extending electromagnetism into relativistic and quantum field theories, where gauge invariance plays a central role.
When charges and currents are time-independent, the electromagnetic fields no longer propagate as waves, and Maxwell’s equations reduce to their static forms:
Electrostatics:∂t∂=0 implies E is time-independent.
Magnetostatics:J(r) is steady, and B is time-independent.
We now derive the integral expressions for V(r) and A(r) in this static regime.
To find the integral form of the vector potential, we play the same trick, but this time, choose to use the Columb gauge. In this gauage, the vector potential equation is
∇2A−μ0ϵ0∂t2∂2A=−μ0J+μ0ϵ0∇∂t∂V.
In the time independent regime however, this reduces to
∇2A=−μ0J,
but this is just Poisson’s equation again, this time for A. It has the solution:
A(r)=4πμ0∫∣r−r′∣J(r′)d3r′
This is the vector potential of a static current distribution in magnetostatics.
These expressions are the static Green’s function solutions to Poisson’s equations for V and A. They form the foundation of the potential formulation in classical electromagnetism.
In this page, we introduced the potential formulation of classical electrodynamics, where the electric and magnetic fields are derived from scalar and vector potentials:
E=−∇V−∂t∂A,B=∇×A
This formulation makes use of the gauge freedom inherent in Maxwell’s equations: the ability to transform the potentials without changing the physical fields. This freedom allows us to impose gauge conditions to simplify calculations.
We explored two common gauge choices:
The Coulomb gauge: ∇⋅A=0, in which the scalar potential satisfies Poisson’s equation and is determined instantaneously by the charge distribution.
The Lorenz gauge: ∇⋅A=−μ0ϵ0∂t∂V, which decouples the potentials and leads to symmetric wave equations for both V and A.
In the static limit, Maxwell’s equations reduce to Poisson equations for the potentials:
These results demonstrate how potentials not only provide an elegant alternative to the field formulation but also reveal deeper mathematical structure and physical symmetries — especially when extended to time-dependent and relativistic electrodynamics.