Expressing Maxwell's equations as scalar equations involving differentials in Euclidean space

Question

I am trying to convert Maxwell's equations from the well known differential form (found on Wikipedia Maxwell's equations) into scalar equations involving partial derivatives (more than four equations). The problem I'm having is finding the definitions of physical quantities. I'm not sure how charge density and current density are expressed in euclidean space. How are these expressed? What are Maxwell's Equations expressed in this way?

Thomas Fritsch · Accepted Answer

I'm not sure how charge density and current density are expressed in euclidean space.

Charge density $rho$ is just a scalar density (measured in Coulomb/m$^3$).
And current density $vec{J}$ is a vector density (with cartesian components
$J_x, J_y, J_z$, measured in Ampere/m$^2$).

What are Maxwell's Equations expressed in this way?

Maxwell's equations (written with vector-calculus) are given by
$$
begin{align}
vec{nabla}cdotvec{B} &= 0  tag{1a} 
vec{nabla}cdotvec{E} &= frac{rho}{epsilon_0} tag{1b} 
vec{nabla}timesvec{E} + frac{partialvec{B}}{partial t} &= vec{0} tag{1c} 
vec{nabla}timesvec{B} - frac{1}{c^2} frac{partialvec{E}}{partial t} &= mu_0vec{J} tag{1d} 
end{align}
$$
Using the definitions of divergence ($vec{nabla}cdot$)
and curl ($vec{nabla}times$) it is straight-forward
to write the above equations with the cartesian field components
($E_x,E_y,E_z,B_x,B_y,B_z$) in cartesian coordinates ($x,y,z$).
The scalar equations (1a) and (1b) each remain one equation.
The vector equations (1c) and (1d) each expand to three equations (for $x, y, z$ components).
$$
begin{align}
frac{partial B_x}{partial x} + frac{partial B_y}{partial y} + frac{partial B_z}{partial z} &= 0 
& 
frac{partial E_x}{partial x} + frac{partial E_y}{partial y} + frac{partial E_z}{partial z} &= frac{rho}{epsilon_0} 
& 
frac{partial E_z}{partial y} - frac{partial E_y}{partial z} + frac{partial B_x}{partial t} &= 0 
frac{partial E_x}{partial z} - frac{partial E_z}{partial x} + frac{partial B_y}{partial t} &= 0 
frac{partial E_y}{partial x} - frac{partial E_x}{partial y} + frac{partial B_z}{partial t} &= 0 
& 
frac{partial B_z}{partial y} - frac{partial B_y}{partial z} - frac{1}{c^2}frac{partial E_x}{partial t} &= mu_0 J_x 
frac{partial B_x}{partial z} - frac{partial B_z}{partial x} - frac{1}{c^2}frac{partial E_y}{partial t} &= mu_0 J_y 
frac{partial B_y}{partial x} - frac{partial B_x}{partial y} - frac{1}{c^2}frac{partial E_z}{partial t} &= mu_0 J_z
end{align}
$$

R. Romero · Answer

If I understand you correctly,there are a couple ways you can proceed.

Method 1:

Maxwell's Equations in vector form already represent partial differential equations of the components of the Electric and Magnetic Fields. You can re-express the standard equations strictly in terms of those components.

This can be done most compactly using Einstein's Summation notation.Repeated indices in a sum are assumed to be summed over even without the usual $Sigma$

In this way, the dot product of two vectora $vec{A}$ and $vec{B}$ is:

$$A_xB_x+A_yB_y+A_zB_z=sum_{i=1}^3A_iB_i=A_iB_i$$

$$nabla cdot vec{E}= frac{partial E_x}{partial x}+frac{partial E_y}{partial y}+frac{partial E_z}{partial z}=sum_{k=1}^3frac{partial}{partial x_i}E_i=frac{partial}{partial x^i}E_i$$

Where $i$ ranges from 1 to 3, $x_1=x, x_2=y, x_3=z$

You drop the sum easing the notation especially when you start using double sums.

In this way you can represent the curl of a vector field component by component:

$$ (nabla times vec{B})_i=epsilon_{ijk}frac{partial}{x_j}B_k$$

Putting it all together with Maxwell's equations, you get:

$$frac{partial}{partial x_i}E_i=rho/epsilon_o$$
$$frac{partial}{partial x_i}B_i=0$$
$$epsilon_{ijk}frac{partial}{partial x_i}E_i=-frac{partial}{partial t}B_i$$
$$epsilon_{ijk}frac{partial}{partial x_i}B_i=mu_0J_i+frac{partial}{partial t}E_i$$

Method 2:

If you let $vec{B}=nabla times vec{A}$ and $vec{E}=-nabla V - frac{partial}{partial}vec{A}$ where $V$ is the scalar potential and $vec{A}$ is the magnetic vector potential, and use the Lorentz Gauage, then you can rewrite Maxwell's equations in the form of 4 (net) equations.

$$vec{E}=-nabla V - partialvec{A}/partial t$$
$$nabla cdot vec{E}=rho/epsilon_0=-nabla^2 V-partial(nabla cdot vec{A})partial t$$

By the Lorentz Gauge, $nabla cdot vec{A} = -frac{1}{c^2}frac{partial}{partial t}V$

Re arranging, we get:

$$nabla ^2 V - frac{1}{c^2}frac{partial ^2}{partial t ^2}V=-rho/epsilon_0$$

There is also:

$$nabla times vec{B}=nabla times (nabla times vec{A})=nabla(nabla cdot vec{A}) - nabla^2vec{A}=mu_0vec{J}+frac{1}{c^2}frac{partial}{partial t}(-nabla V - partial vec{A}/partial t)$$

rearranging:

$$ nabla(nabla cdot vec{A} + frac{1}{c^2}partial V/ partial t)-nabla^2vec{A}+frac{1}{c^2}frac{partial ^2}{partial t^2}vec{A}= mu_0vec{J}$$

Where the far left term vanishes again the lorentz gauge. Then as in method one, this final equation can be represented component by component is talked about in method one.

Expressing Maxwell's equations as scalar equations involving differentials in Euclidean space

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