\(\cfrac { d\, }{ d\, t } \oint { E_{\psi} } \, d\, A_{ l }+\cfrac { 1 }{\varepsilon _{ new }}\oint{ J_{\psi}}\,d\,A_l=\oint _{ 2\pi a_{ \psi } }{ F_{ \psi } } \, d\, \lambda\)
Replacing \(\varepsilon _{ new }\) with \(\varepsilon _{ o}\) and multiplying by \(\mu_{o}\varepsilon_{o}\),
\(\mu_{o}\varepsilon_{o}\cfrac { d\, }{ d\, t } \oint { E_{\psi} } \, d\, A_{ l }+\mu_{o}\oint{ J_{\psi}}\,d\,A_l=\mu_{o}\varepsilon_{o}\oint _{ 2\pi a_{ \psi } }{ F_{ \psi } } \, d\, \lambda\)
\(\mu_{o}\varepsilon_{o}\cfrac { d\, }{ d\, t } \oint { E_{\psi} } \, d\, A_{ l }+\mu_{o}\oint{ J_{\psi}}\,d\,A_l=\cfrac{1}{c^2}\oint _{ 2\pi a_{ \psi } }{ F_{ \psi } } \, d\, \lambda\)
Since \(a_{\psi}\) is the radius of a circle, when we rotate \(A_l\) by \(2\pi\) along any axis of symmetry through \(A_l\), we see that \(E_{\psi}\) cancels with \(-E_{\psi}\) when \(A_l\) has been rotated by \(\pi\). Similarly, \(J_{\psi}\) cancels with \(-J_{\psi}\) at the rotation of \(\pi\). When so rotated, the integral \(\oint_{2\pi a_\psi}{F_{\psi}}\,d\,\lambda\) describes a closed surface area containing the volume defined by \(A_l\) after \(2\pi\) rotation. As such,
\(\require{cancel}\)
\(\cancelto{0}{\mu_{o}\varepsilon_{o}\cfrac { d\, }{ d\, t } \oint { E_{\psi} } \, d\,V}+\cancelto{0}{\mu_{o}\oint{ J_{\psi}}\,d\,V}=\cfrac{1}{c^2}\oint _{ 4\pi a^2_{ \psi } }{ F_{ \psi } } \, d\, A_s=0\)
This is the second Maxwell's equation, Gauss's law for magnetism.
Consider the fourth Maxwell equation again, if we set \(J_\psi=0\) and integrate along \(a_{\psi}\), the radius of the circle defined by the perimeter \(A_l\),
\(\require{cancel}\)
\(\mu _{ o }\varepsilon _{ o }\int { \cfrac { d\, }{ d\, t } \oint { E_{ \psi } } \, d\, A_{ l } } dx+\mu _{ o }\int { \oint {\cancelto{0}{ J_{ \psi }} } \, d\, A_{ l } } dx=\cfrac { 1 }{ c^{ 2 } } \int { \oint _{ 2\pi a_{ \psi } }{ F_{ \psi } } \, d\, \lambda } dx\)
The loop \(2\pi a_\psi\) becomes an area \(A_l=\pi a^2_{\psi}\), therefore,
\(\mu _{ o }\varepsilon _{ o }\int { \cfrac { d\, }{ d\, t } \oint { E_{ \psi } } \, d\, A_{ l } } dx=\cfrac { 1 }{ c^{ 2 } } { \oint _{ \pi a^2_{ \psi } }{ F_{ \psi } } \, d\, A_l } \)
It is totally arbitrary that we move at speed \(c\), and since \(x\) is perpendicular to \(\lambda\) and \(A_l\).
Using \(A_l\) are the reference direction
\( \cfrac { dx }{ dt } =ic\)
\( \mu _{ o }\varepsilon _{ o }ic\int { \cfrac { d\, }{ d\, t } \oint { E_{ \psi } } \, d\, A_{ l } } dt=\cfrac { 1 }{ c^{ 2 } } \oint _{\pi a^2_{\psi}}{ F_{ \psi } } dA_{ l }\)
\( c=\cfrac { 1 }{ \sqrt { \mu _{ o }\varepsilon _{ o } } } \)
\( ic\oint { E_{ \psi } } \, d\, A_{ l }= \oint_{\pi a^2_{\psi}} { F_{ \psi } } dA_{ l }\)
In a similar way, we consider \(A_l\) as the integral of a loop \(2\pi a_{\psi}\) outwards along \(dx\). This loop is obtained by integrating along \(d\lambda\) for \(2\pi a_{\psi}\).
\( ic\int\oint _{2\pi a_\psi}{ E_{ \psi } } \, d\, \lambda dx= \oint _{\pi a^2_{\psi}} { F_{ \psi } } dA_{ l }\)
Similarly, \( \cfrac { dx }{ dt } =ic\)
\( (ic)^2\int\oint_{2\pi a_\psi} { E_{ \psi } } \, d\, \lambda d\,t= \oint _{\pi a^2_{\psi}} { F_{ \psi } } dA_{ l }\)
\( \oint_{2\pi a_\psi} { E_{ \psi } } \, d\, \lambda =-\cfrac{1}{c^2} \cfrac{d}{d\,t}\left\{\oint _{\pi a^2_{\psi}}{ F_{ \psi } } dA_{ l }\right\}\)
This is the third Maxwell's Equation. Consider again the fourth Maxwell equation, when we set \(F_\psi=0\)
\(\mu _{ o }\varepsilon _{ o }\cfrac { d\, }{ d\, t } \oint_{\pi a^2_\psi} { E_{ \psi }} \, d\, A_{ l }+\mu _{ o }\oint _{\pi a^2_\psi}{ J_{ \psi } } \, d\, A_{ l }=\cfrac { 1 }{ c^{ 2 } } \oint _{ 2\pi a_{ \psi } }{ \cancelto{0}{F_{ \psi } }} \, d\, \lambda \)
If we consider \(A_l\) to be from the integration of \(2\pi a_{\psi}\) along the radius \(x\), using \(A_l\) as reference,
\(i\lambda=A_l\) and \(\cfrac{dx}{d\,t}=ic\)
\(\varepsilon _{ o }\cfrac { d\, }{ d\, t }\int{ \oint _{ 2\pi a_{ \psi } } { E_{ \psi }}} \, d\, i\lambda d\,x+\oint _{\pi a^2_\psi}{ J_{ \psi } } \, d\, A_{ l }=0 \)
\(\varepsilon _{ o }(i)^2c\cfrac { d\, }{ d\, t }\int{ \oint _{ 2\pi a_{ \psi } }{ E_{ \psi }}} \, d\, \lambda d\,t+\oint _{\pi a^2_\psi}{ J_{ \psi } } \, d\, A_{ l }=0 \)
\(\varepsilon _{ o }c\oint _{ 2\pi a_{ \psi } } { E_{ \psi }} \, d\, \lambda=\oint _{\pi a^2_\psi}{ J_{ \psi } } \, d\, A_{ l } \)
\(J_{ \psi }=\cfrac { 1 }{ 2\pi a^{ 2 }_{ \psi } } \cfrac { d\, q_{ \psi } }{ d\, t } \)
Substitute in \(J_{\psi}\) and integrate over time, t.
\(\varepsilon _{ o }c\int { \oint _{ 2\pi a_{ \psi } } { E_{ \psi } } } \, d\, \lambda d\, t=\oint _{ \pi a^{ 2 }_{ \psi } }{ \int { \cfrac { 1 }{ 2\pi a^{ 2 }_{ \psi } } \cfrac { d\, q_{ \psi } }{ d\, t } } } d\, t\, d\, A_{ l }\)
\(E_\psi\) is constant,
\(\varepsilon _{ o }ct\oint _{ 2\pi a_{ \psi } }{ E_{ \psi } } d\, \lambda =\oint _{ \pi a^{ 2 }_{ \psi } }{ \cfrac { q }{ 2\pi a^{ 2 }_{ \psi } } } \, d\, A_{ l }\)
\(\varepsilon _{ o }\oint _{ 2\pi a_{ \psi } }{ E_{ \psi } } d\, \lambda =\oint _{ \pi a^{ 2 }_{ \psi } }{ \cfrac { q }{ 2\pi a^{ 2 }_{ \psi }ct } } \, d\, A_{ l }\)
where \(vol=2\pi a^{ 2 }_{ \psi }ct\) is the volume transcribed by \(J\) with a base of \(2\pi a^{ 2 }_{ \psi }\) over time \(t\).
With \(t\) small,
\(\rho=\cfrac { q_\psi }{ 2\pi a^{ 2 }_{ \psi }ct } \)
where \(\rho\) is the volume charge density. So,
\(\varepsilon _{ o }\oint_{2\pi a_{\psi}} { E_{ \psi } } d\, \lambda =\oint _{ \pi a^{ 2 }_{ \psi } }{ \rho } \, d\, A_{ l }\)
We then rotate \(2\pi a_{\psi}\) into a surface containing the volume define by \(A_l\) as \(A_l\) rotates correspondingly,
\(\oint_{4\pi a^2_{\psi}} { E_{ \psi } } d\, A_l=\cfrac{1}{\varepsilon _{ o }}\oint { \rho } \, d\,V\)
This is the first Maxwell's Equation, Gauss's Law.
Thanks to the fact that \(a_{\psi}\) describes circles and spheres, we have all four Maxwell's Equations for particles. \(E_\psi\) is the field around the particle, \(F_\psi\) is a field analogous to the \(B\) field of charges, produce by the particle in motion.
