Weber Electrodynamics

Check Ampere's Law In The Coulomb Gauge

We wish to verify Ampere's Law, one of Maxwell's four differential equations, when the Coulomb gauge condition is imposed. As we have demonstrated on other web pages, the Coulomb gauge condition is traditionally imposed to make Gauss's Law (another of Maxwell's equations) correct. The question is: Does the imposition of the Coulomb gauge condition make all of Maxwell's equations mathematically correct?

We are working in the non-special relativity regime, and where time retardation can be ignored. We use the SI system of units.

First, we will distinguish between the coordinates of the sources by using a subscript "S", and the coordinates of the detector (observation, or test) position by using a subscript "T". Then, for example, the \( \nabla \) vector operator, when being applied to the detector position, will be written as, in a Cartesian coordinate system $$ \nabla_T \equiv \dfrac{\partial}{\partial x_T}\hat{x} + \dfrac{\partial}{\partial y_T}\hat{y} + \dfrac{\partial}{\partial z_T}\hat{z}. $$

We use the scalar $$ \Phi \equiv \dfrac{1}{4 \pi \epsilon_0} \iiint \rho \dfrac{1}{r_{TS}} d\tau_S $$ and the vector $$ \vec{A} \equiv \dfrac{1}{4 \pi \epsilon_0 c^2} \iiint \vec{J} \dfrac{1}{r_{TS}} d\tau_S $$ potential function definitions, where \( r_{TS} \) is the distance from the differential source element \( d\tau_S \) to the detector (observation, or test) position, \( \rho \) is the charge density function, and \( \vec{J} \) is the current density function defined by $$ \vec{J} \equiv \rho \vec{v}_S $$ and \( \vec{v}_S \) is the "average drift velocity" of the source charges passing through the differential volume element \( d\tau_S .\)

The eletric field can be decomposed into the gradient of the scalar potential function and the time derivative of the vector potential function as $$ \vec{E} = -\nabla_T \Phi - \dfrac{\partial}{\partial t} \vec{A}. $$

The magnetic induction field is defined by $$ \vec{B} = \nabla_T \times \vec{A}. $$

We have seen on other web pages that the Maxwell equation known as Gauss's Law $$ \nabla_T \bullet \vec{E} = \dfrac{1}{\epsilon_0} \rho $$ does not hold in general for the above scalar and vector potential functions.

However, in the special case that the Coulomb gauge condition $$ \nabla_T \bullet \vec{A} = 0 $$ is applied, Gauss's Law then becomes mathematically verifiable.

Perhaps all of Maxwell's equations, with the above definitions of the scalar and vector potential functions, are then mathematically correct in this Coulomb gauge? We need to check this. In particular, we check that Ampere's Law $$ \nabla_T \times \vec{B} = \mu_0 \vec{J}_f + \mu_0 \epsilon_0 \dfrac{\partial}{\partial t} \vec{E} $$ is (or isn't) mathematically correct when using the Coulomb gauge condition.

Using the definition of the magnetic induction field in terms of the vector potential function, given above, we can write the left hand side of Ampere's Law as $$ \nabla_T \times \vec{B} = \nabla_T \times (\nabla_T \times \vec{A}). $$

Using the relation $$ \nabla_T \times (\nabla_T \times \vec{A}) = \nabla_T (\nabla_T \bullet \vec{A}) - \nabla_T^2 \vec{A} $$ we can then use the Coulomb gauge condition $$ \nabla_T \bullet \vec{A} = 0 $$ to write $$ \nabla_T \times \vec{B} = - \nabla_T^2 \vec{A}. $$

Now, inserting the definition of the vector potential function, we have $$ \nabla_T \times \vec{B} = - \nabla_T^2 \left( \dfrac{1}{4 \pi \epsilon_0 c^2} \iiint \vec{J} \dfrac{1}{r_{TS}} d\tau_S \right) $$

Since the integration is over the source coordinates ("S" labeled parameters) while the spatial differentiation is over the test coordinates ("T" labeled parameters), we can bring the Laplacian operator under the integration. Further, we note that the source current density function is not a function of the test coordinates, so we can write $$ \nabla_T \times \vec{B} = - \dfrac{1}{4 \pi \epsilon_0 c^2} \iiint \vec{J} \nabla_T^2 \dfrac{1}{r_{TS}} d\tau_S . $$

We now use the well known relation $$ \nabla_T^2 \dfrac{1}{r_{TS}} = -4 \pi \delta(\vec{r}_T - \vec{r}_S) $$ to write $$ \nabla_T \times \vec{B} = - \dfrac{1}{4 \pi \epsilon_0 c^2} \iiint \vec{J} \left( -4 \pi \delta(\vec{r}_T - \vec{r}_S) \right) d\tau_S . $$

Pulling out the constants $$ \nabla_T \times \vec{B} = \dfrac{1}{\epsilon_0 c^2} \iiint \vec{J} \delta(\vec{r}_T - \vec{r}_S) d\tau_S $$ and performing the integration gives $$ \nabla_T \times \vec{B} = \dfrac{1}{\epsilon_0 c^2} \vec{J}_f $$ which is the same as $$ \nabla_T \times \vec{B} = \mu_0 \vec{J}_f $$ where $$ \vec{J}_f \equiv \vec{J}(\vec{r}_T, t) $$ is called the "free current density function".

It is obvious now that this result is not the same as Ampere's Law, repeated here: $$ \nabla_T \times \vec{B} = \mu_0 \vec{J}_f + \mu_0 \epsilon_0 \dfrac{\partial}{\partial t} \vec{E}. $$

Therefore, not all of Maxwell's equations are mathematically correct when using the above defined scalar and vector potential functions and the Coulomb gauge condition.

Is there any gauge for which the Maxwell equations are mathematically verifiable using these scalar and vector potential functions?

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