Maxwell Equations

Equation of Continuity

\[\begin{align} & {{i}_{entering}}=-{{i}_{leaving}} \\ & \Rightarrow \frac{\partial q}{\partial t}=-J\cdot d\vec{S} \\ & \Rightarrow \frac{\partial q}{\partial t}+J\cdot d\vec{S}=0 \\ & \Rightarrow \frac{\partial \rho dV}{\partial t}+J\cdot d\vec{S}=0 \\ & \Rightarrow \frac{\partial \int{\rho dV}}{\partial t}+\int{\left( \nabla \cdot \vec{J} \right)dV}=0 \\ & \Rightarrow \int{\frac{\partial \rho }{\partial t}dV}+\int{\left( \nabla \cdot \vec{J} \right)dV}=0 \\ & \Rightarrow \left( \nabla \cdot \vec{J} \right)+\frac{\partial \rho }{\partial t}=0 \\ \end{align}\]

Derivation of Maxwell's first field eqaution

\[\begin{align} & \oint\limits_{S}{\vec{E}\cdot d\vec{S}}=\phi \\ & \Rightarrow \oint\limits_{S}{\vec{E}\cdot d\vec{S}}=\frac{q}{{{\varepsilon }_{0}}} \\ & \Rightarrow \oint\limits_{S}{\vec{E}\cdot d\vec{S}}=\frac{\int{dq}}{{{\varepsilon }_{0}}} \\ & \Rightarrow \oint\limits_{S}{\vec{E}\cdot d\vec{S}}=\frac{\int{\rho dV}}{{{\varepsilon }_{0}}} \\ & \Rightarrow \int\limits_{V}{\left( \nabla \cdot \vec{E} \right)}\,dV=\frac{\int{\rho dV}}{{{\varepsilon }_{0}}} \\ & \Rightarrow \nabla \cdot \vec{E}=\frac{\rho }{{{\varepsilon }_{0}}} \\ \end{align}\]

Derivation of Maxwell 2nd Law

Since the Magnetic flux through any surface does not exist

\[\begin{align} & \oint\limits_{S}{\vec{B}\cdot d\vec{S}}=\phi =0 \\ & \Rightarrow \int\limits_{V}{\left( \nabla \cdot \vec{B} \right)}\,dV=0 \\ & \Rightarrow \nabla \cdot \vec{B}=0 \\ \end{align}\]

Derivation of Maxwells 3rd Law

\[\begin{align} & \int\limits_{l}{\vec{E}\cdot d\vec{l}}=-\frac{\partial \phi }{\partial t} \\ & \int\limits_{S}{\left( \nabla \times \vec{E} \right)\cdot dS}=-\frac{\partial }{\partial t}\left( \int\limits_{S}{\vec{B}\cdot d\vec{S}} \right) \\ & \Rightarrow \int\limits_{S}{\left( \nabla \times \vec{E} \right)\cdot dS}=-\left( \int\limits_{S}{\frac{\partial \vec{B}}{\partial t}\cdot d\vec{S}} \right) \\ & \Rightarrow \nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t} \\ \end{align}\]

Maxwell's 4th equation

\[\begin{align} & \int\limits_{l}{\vec{B}\cdot d\vec{l}}={{\mu }_{0}}{{i}_{enclosed}} \\ & \Rightarrow \int\limits_{l}{\vec{B}\cdot d\vec{l}}={{\mu }_{0}}\int\limits_{S}{\vec{J}\cdot d\vec{S}} \\ & \Rightarrow \int\limits_{S}{\left( \nabla \times \vec{B} \right)\cdot d\vec{S}}=\int\limits_{S}{{{\mu }_{0}}\vec{J}\cdot d\vec{S}} \\ & \Rightarrow \nabla \times \vec{B}={{\mu }_{0}}\vec{J} \\ \end{align}\]

If we take divergence on both sides \[\begin{align} & \Rightarrow \nabla \cdot \left( \nabla \times \vec{B} \right)={{\mu }_{0}}\left( \nabla \cdot \vec{J} \right) \\ & \Rightarrow 0={{\mu }_{0}}\left( \nabla \cdot \vec{J} \right) \\ & \Rightarrow \nabla \cdot \vec{J}=0 \\ & \because \frac{\partial \rho }{\partial t}+\nabla \cdot \vec{J}=0 \\ & \Rightarrow -\frac{\partial \rho }{\partial t}=0 \\ & \Rightarrow \rho \text{ is either zero or constant }\text{.} \\ & \text{Hence charge is zero or independent of time} \\ \end{align}\]

Since Electric field can be space and time dependent , Maxwell Modified Ampere's Law to include another current called displacement current which exists in insulators due to changing electric field. Hence , \[\begin{align} & \vec{J}={{{\vec{J}}}_{C}}+{{{\vec{J}}}_{D}} \\ & \Rightarrow \nabla \times \vec{B}={{\mu }_{0}}\left( {\vec{J}} \right) \\ & \Rightarrow \nabla \times \vec{B}={{\mu }_{0}}\left( {{{\vec{J}}}_{C}}+{{{\vec{J}}}_{D}} \right) \\ & \Rightarrow \nabla \times \vec{B}={{\mu }_{0}}{{{\vec{J}}}_{C}}+{{\mu }_{0}}{{{\vec{J}}}_{D}} \\ & \Rightarrow \nabla \cdot \left( \nabla \times \vec{B} \right)={{\mu }_{0}}\left( \nabla \cdot {{{\vec{J}}}_{C}} \right)+{{\mu }_{0}}\left( \nabla \cdot {{{\vec{J}}}_{D}} \right) \\ & \Rightarrow 0={{\mu }_{0}}\left( -\frac{\partial \rho }{\partial t} \right)+{{\mu }_{0}}\left( \nabla \cdot {{{\vec{J}}}_{D}} \right) \\ & \Rightarrow \nabla \cdot {{{\vec{J}}}_{D}}=\frac{\partial \rho }{\partial t} \\ & =\frac{\partial }{\partial t}\left( {{\varepsilon }_{0}}\left( \nabla \cdot \vec{E} \right) \right) \\ & =\frac{\partial }{\partial t}\left( \nabla \cdot {{\varepsilon }_{0}}\vec{E} \right) \\ & =\nabla \cdot {{\varepsilon }_{0}}\frac{\partial \vec{E}}{\partial t} \\ & \Rightarrow {{{\vec{J}}}_{D}}={{\varepsilon }_{0}}\frac{\partial \vec{E}}{\partial t} \\ \end{align}\]

Alternative method to find Displacement Current

\[\begin{align} & {{J}_{D}}=\frac{i}{A} \\ & =\frac{\frac{\partial q}{\partial t}}{A} \\ & =\frac{1}{A}\frac{\partial q}{\partial t} \\ & =\frac{1}{A}\frac{\partial CV}{\partial t} \\ & =\frac{1}{A}\frac{{{\varepsilon }_{0}}A}{d}\frac{\partial V}{\partial t} \\ & ={{\varepsilon }_{0}}\frac{\partial }{\partial t}\left( \frac{V}{d} \right) \\ & \therefore {{J}_{D}}={{\varepsilon }_{0}}\frac{\partial E}{\partial t} \\ \end{align}\]

Gauge Transformation conditions

\[\begin{align} & {{\phi }_{2}}={{\phi }_{1}}+\alpha \\ & {{{\vec{A}}}_{2}}={{{\vec{A}}}_{1}}+\beta \\ & \because {{E}_{2}}={{E}_{1}} \\ & \Rightarrow -\nabla {{\phi }_{2}}-\frac{\partial {{A}_{2}}}{\partial t}=-\nabla {{\phi }_{1}}-\frac{\partial {{A}_{1}}}{\partial t} \\ & \Rightarrow -\nabla \left( {{\phi }_{1}}+\alpha \right)-\frac{\partial }{\partial t}\left( {{A}_{1}}+\beta \right)=-\nabla {{\phi }_{1}}-\frac{\partial {{A}_{1}}}{\partial t} \\ & \Rightarrow -\nabla {{\phi }_{1}}-\frac{\partial {{A}_{1}}}{\partial t}-\nabla \alpha -\frac{\partial \beta }{\partial t}=-\nabla {{\phi }_{1}}-\frac{\partial {{A}_{1}}}{\partial t} \\ & \Rightarrow -\nabla \alpha -\frac{\partial \beta }{\partial t}=0 \\ & \Rightarrow \nabla \alpha =-\frac{\partial \beta }{\partial t} \\ & \because {{{\vec{B}}}_{2}}={{{\vec{B}}}_{1}} \\ & \Rightarrow \nabla \times {{{\vec{A}}}_{2}}=\nabla \times {{{\vec{A}}}_{1}} \\ & \Rightarrow \nabla \times \left( {{{\vec{A}}}_{1}}+\beta \right)=\nabla \times {{{\vec{A}}}_{1}} \\ & \Rightarrow \nabla \times \beta =0 \\ & \Rightarrow \beta =-\nabla \lambda \\ & \Rightarrow \nabla \alpha =-\frac{\partial \beta }{\partial t} \\ & \Rightarrow \nabla \alpha =-\frac{\partial }{\partial t}\left( -\nabla \lambda \right) \\ & \Rightarrow \alpha =\frac{\partial \lambda }{\partial t} \\ & \therefore {{\phi }_{2}}={{\phi }_{1}}+\frac{\partial \lambda }{\partial t} \\ & {{{\vec{A}}}_{2}}={{{\vec{A}}}_{1}}-\nabla \lambda \\ \end{align}\]

Wave Equations in terms of Electric field

\[\begin{align} & \nabla \times \left( \nabla \times \vec{E} \right)=-\nabla \times \frac{\partial \vec{B}}{\partial t} \\ & \Rightarrow \nabla \left( \nabla \cdot \vec{E} \right)-\left( \nabla \cdot \nabla \right)\vec{E}=-\frac{\partial }{\partial t}\left( \nabla \times \vec{B} \right) \\ & \Rightarrow \nabla \left( \frac{\rho }{{{\varepsilon }_{0}}} \right)-{{\nabla }^{2}}\vec{E}=-\frac{\partial }{\partial t}\left( {{\mu }_{0}}\vec{J}+{{\mu }_{0}}{{\varepsilon }_{0}}\frac{\partial E}{\partial t} \right)\left( \because \nabla \cdot \vec{E}=\frac{\rho }{{{\varepsilon }_{0}}} \right) \\ & \Rightarrow 0-{{\nabla }^{2}}\vec{E}=-{{\mu }_{0}}\frac{\partial \vec{J}}{\partial t}-{{\mu }_{0}}{{\varepsilon }_{0}}\frac{{{\partial }^{2}}\vec{E}}{\partial {{t}^{2}}} \\ & \Rightarrow {{\nabla }^{2}}\vec{E}-{{\mu }_{0}}{{\varepsilon }_{0}}\frac{{{\partial }^{2}}\vec{E}}{\partial {{t}^{2}}}-{{\mu }_{0}}\frac{\partial \vec{J}}{\partial t}=0 \\ & \Rightarrow {{\nabla }^{2}}\vec{E}-{{\mu }_{0}}{{\varepsilon }_{0}}\frac{{{\partial }^{2}}\vec{E}}{\partial {{t}^{2}}}-{{\mu }_{0}}\sigma \frac{\partial \vec{E}}{\partial t}=0 \\ \end{align}\]

Wave Equation in terms of Magnetic Field

Wave Equation in terms of Electromagnetic Potentials

\[\begin{align} & \nabla \cdot \vec{E}=\frac{\rho }{{{\varepsilon }_{0}}} \\ & \Rightarrow \nabla \cdot \left( -\nabla \phi -\frac{\partial \vec{A}}{\partial t} \right)=\frac{\rho }{{{\varepsilon }_{0}}} \\ & \Rightarrow -{{\nabla }^{2}}\phi -\frac{\partial }{\partial t}\left( \nabla \cdot \vec{A} \right)=\frac{\rho }{{{\varepsilon }_{0}}} \\ & \left( \nabla \cdot \vec{A} \right)=0\,\,\,\,\,\,\left( \text{Coulomb Condition} \right) \\ & \Rightarrow {{\nabla }^{2}}\phi =-\frac{\rho }{{{\varepsilon }_{0}}} \\ \end{align}\]

Wave equation in terms of \( \phi\) and \( A\)

\[\begin{align} & \nabla \times \vec{B}={{\mu }_{0}}\vec{J}+{{\mu }_{0}}{{\varepsilon }_{0}}\frac{\partial \vec{E}}{\partial t} \\ & \Rightarrow \nabla \times \left( \nabla \times \vec{A} \right)={{\mu }_{0}}\vec{J}+{{\mu }_{0}}{{\varepsilon }_{0}}\frac{\partial \vec{E}}{\partial t} \\ & \Rightarrow \nabla \left( \nabla \cdot \vec{A} \right)-{{\nabla }^{2}}\vec{A}-{{\mu }_{0}}{{\varepsilon }_{0}}\frac{\partial \vec{E}}{\partial t}={{\mu }_{0}}\vec{J} \\ & \Rightarrow \nabla \left( \nabla \cdot \vec{A} \right)-{{\nabla }^{2}}\vec{A}-{{\mu }_{0}}{{\varepsilon }_{0}}\frac{\partial }{\partial t}\left( -\nabla \phi -\frac{\partial \vec{A}}{\partial t} \right)={{\mu }_{0}}\vec{J} \\ & \Rightarrow \nabla \left( \nabla \cdot \vec{A} \right)-{{\nabla }^{2}}\vec{A}+\nabla \left( {{\mu }_{0}}{{\varepsilon }_{0}}\frac{\partial \phi }{\partial t} \right)+{{\mu }_{0}}{{\varepsilon }_{0}}\frac{{{\partial }^{2}}\vec{A}}{\partial {{t}^{2}}}={{\mu }_{0}}\vec{J} \\ & \Rightarrow -{{\nabla }^{2}}\vec{A}+{{\mu }_{0}}{{\varepsilon }_{0}}\frac{{{\partial }^{2}}\vec{A}}{\partial {{t}^{2}}}+\nabla \left( \nabla \cdot \vec{A}+{{\mu }_{0}}{{\varepsilon }_{0}}\frac{\partial \phi }{\partial t} \right)={{\mu }_{0}}\vec{J} \\ & \nabla \cdot \vec{A}+{{\mu }_{0}}{{\varepsilon }_{0}}\frac{\partial \phi }{\partial t}\left( \text{Lorentz}\,\text{Conditin} \right) \\ & \Rightarrow -{{\nabla }^{2}}\vec{A}+{{\mu }_{0}}{{\varepsilon }_{0}}\frac{{{\partial }^{2}}\vec{A}}{\partial {{t}^{2}}}={{\mu }_{0}}\vec{J} \\ & \Rightarrow {{\nabla }^{2}}\vec{A}-{{\mu }_{0}}{{\varepsilon }_{0}}\frac{{{\partial }^{2}}\vec{A}}{\partial {{t}^{2}}}=-{{\mu }_{0}}\vec{J} \\ \end{align}\] \[\begin{align} & -{{\nabla }^{2}}\phi -\frac{\partial }{\partial t}\left( -{{\mu }_{0}}{{\varepsilon }_{0}}\frac{\partial \phi }{\partial t} \right)=\frac{\rho }{{{\varepsilon }_{0}}} \\ & \Rightarrow -{{\nabla }^{2}}\phi +{{\mu }_{0}}{{\varepsilon }_{0}}\frac{{{\partial }^{2}}\phi }{\partial {{t}^{2}}}=\frac{\rho }{{{\varepsilon }_{0}}} \\ & \Rightarrow {{\nabla }^{2}}\phi -{{\mu }_{0}}{{\varepsilon }_{0}}\frac{{{\partial }^{2}}\phi }{\partial {{t}^{2}}}=\frac{\rho }{{{\varepsilon }_{0}}} \\ \end{align}\]

Special Cases

When \( \rho \) is zero
When \( \rho \) is zero

Pyonting Theorem

Poynting Theorem Explains How Electromagnetic Energy Travels Through Space. It Relates changes in Energy Density to the Poynting Vector, which indicates the direction and Amount of Energy transferred in the Electromagnetic fields. The rate of energy transfer (per unit volume) from a region of space equals the rate of workdone on a charge distribution plus the energy flux leaving that region. Another statement : - "The decrease in the electromagnetic energy per unit time in a certain volume is equal to the sum of work done by the field forces and the net outward flux per unit time". The net power flowing out of a given volume v is equal to the time rate of decrease in the energy stored within volume v minus the ohmic power dissipated. \[\begin{align} dW &=\vec{F}\cdot d\vec{l} \\ & =dq\vec{E}\cdot d\vec{l} \\ & =dq\vec{E}\cdot \vec{v}dt \\ & =\rho dV\vec{E}\cdot \vec{v}dt \\ & =\left( \vec{E}\cdot \rho \vec{v} \right)dVdt \\ & =\left( \vec{E}\cdot \vec{J} \right)dVdt \\ & \frac{dW}{dt}=\left( \vec{E}\cdot \vec{J} \right)dV \\ & =\left( \vec{E}\cdot \left( \frac{\nabla \times \vec{B}-{{\mu }_{0}}{{\varepsilon }_{0}}\frac{\partial \vec{E}}{\partial t}}{{{\mu }_{0}}} \right) \right)dV \\ & =\left( \frac{\vec{E}\cdot \left( \nabla \times \vec{B} \right)-{{\mu }_{0}}{{\varepsilon }_{0}}\left( \vec{E}\cdot \frac{\partial \vec{E}}{\partial t} \right)}{{{\mu }_{0}}} \right)dV \\ & =\left( \frac{-\nabla \cdot \left( \vec{E}\times \vec{B} \right)+\vec{B}\cdot \left( \nabla \times \vec{E} \right)-{{\mu }_{0}}{{\varepsilon }_{0}}\left( \vec{E}\cdot \frac{\partial \vec{E}}{\partial t} \right)}{{{\mu }_{0}}} \right)dV \\ & =\left( \frac{-\nabla \cdot \left( \vec{E}\times \vec{B} \right)-\vec{B}\cdot \frac{\partial \vec{B}}{\partial t}-{{\mu }_{0}}{{\varepsilon }_{0}}\left( \vec{E}\cdot \frac{\partial \vec{E}}{\partial t} \right)}{{{\mu }_{0}}} \right)dV \\ & =\left( -{{\varepsilon }_{0}}\left( \vec{E}\cdot \frac{\partial \vec{E}}{\partial t} \right)-\frac{1}{{{\mu }_{0}}}\vec{B}\cdot \frac{\partial \vec{B}}{\partial t}-\frac{\nabla \cdot \left( \vec{E}\times \vec{B} \right)}{{{\mu }_{0}}} \right)dV \\ & =\left( -\frac{{{\varepsilon }_{0}}}{2}\left( \vec{E}\cdot \frac{\partial \vec{E}}{\partial t}+\frac{\partial E}{\partial t}\cdot \vec{E} \right)-\frac{1}{{{\mu }_{0}}}\vec{B}\cdot \frac{\partial \vec{B}}{\partial t}-\frac{\nabla \cdot \left( \vec{E}\times \vec{B} \right)}{{{\mu }_{0}}} \right)dV \\ & =\left( -\frac{{{\varepsilon }_{0}}}{2}\frac{\partial }{\partial t}\left( \vec{E}\cdot \vec{E} \right)-\frac{1}{{{\mu }_{0}}}\frac{1}{2}\frac{\partial }{\partial t}\left( \vec{B}\cdot \vec{B} \right)-\frac{\nabla \cdot \left( \vec{E}\times \vec{B} \right)}{{{\mu }_{0}}} \right)dV \\ & =\left( -\frac{\partial }{\partial t}\left( \frac{1}{2}{{\varepsilon }_{0}}{{E}^{2}}+\frac{{{B}^{2}}}{2{{\mu }_{0}}} \right)-\frac{\nabla \cdot \left( \vec{E}\times \vec{B} \right)}{{{\mu }_{0}}} \right)dV \\ & =-\frac{\partial }{\partial t}\left( \frac{1}{2}{{\varepsilon }_{0}}{{E}^{2}}+\frac{{{B}^{2}}}{2{{\mu }_{0}}} \right)dV-\frac{\nabla \cdot \left( \vec{E}\times \vec{B} \right)}{{{\mu }_{0}}}dV \\ & =-\frac{\partial {{u}_{em}}}{\partial t}dV-\nabla \cdot \vec{S}\,dV \\ & =-\frac{\partial {{U}_{em}}}{\partial t}-\int\limits_{S}{\vec{S}\cdot dA} \\ \end{align}\]

Skin depth of EM wave

It is the depth in the medium after which the amplitude of the electromagnetic wave reduces to 1/e times

\[\begin{align} & {{\nabla }^{2}}E-{{\mu }_{0}}{{\varepsilon }_{0}}\frac{{{\partial }^{2}}E}{\partial {{t}^{2}}}-{{\mu }_{0}}\sigma \frac{\partial E}{\partial t}=0 \\ & \Rightarrow \left( -{{k}^{2}} \right)E-{{\mu }_{0}}{{\varepsilon }_{0}}\left( -{{\omega }^{2}} \right)E-{{\mu }_{0}}\sigma \left( -i\omega \right)E=0 \\ & \Rightarrow -{{k}^{2}}E+{{\mu }_{0}}{{\varepsilon }_{0}}{{\omega }^{2}}E+i{{\mu }_{0}}\sigma \omega E=0 \\ & \Rightarrow -{{k}^{2}}E+{{\mu }_{0}}{{\varepsilon }_{0}}{{\omega }^{2}}E+i{{\mu }_{0}}\sigma \omega E=0 \\ & \Rightarrow {{k}^{2}}E={{\mu }_{0}}{{\varepsilon }_{0}}{{\omega }^{2}}E+i{{\mu }_{0}}\sigma \omega E \\ & \Rightarrow {{k}^{2}}E={{\mu }_{0}}{{\varepsilon }_{0}}{{\omega }^{2}}E\left( 1+i\frac{\sigma }{{{\varepsilon }_{0}}\omega } \right) \\ & \Rightarrow {{k}^{2}}={{\mu }_{0}}{{\varepsilon }_{0}}{{\omega }^{2}}\left( 1+i\frac{\sigma E}{{{\varepsilon }_{0}}\omega E} \right) \\ & \Rightarrow k=\sqrt{{{\mu }_{0}}{{\varepsilon }_{0}}}\omega {{\left( 1+i\frac{\sigma E}{{{\varepsilon }_{0}}\omega E} \right)}^{1/2}} \\ & =\sqrt{{{\mu }_{0}}{{\varepsilon }_{0}}}\omega {{\left( 1+i\frac{{{J}_{C}}}{{{\varepsilon }_{0}}\left( -i\omega \right)E} \right)}^{1/2}} \\ & =\sqrt{{{\mu }_{0}}{{\varepsilon }_{0}}}\omega {{\left( 1+i\frac{{{J}_{C}}}{{{\varepsilon }_{0}}\frac{\partial E}{\partial t}} \right)}^{1/2}} \\ & =\sqrt{{{\mu }_{0}}{{\varepsilon }_{0}}}\omega {{\left( 1+i\frac{{{J}_{C}}}{\left| {{J}_{D}} \right|} \right)}^{1/2}} \\ & {{J}_{C}}>>>>\left| {{J}_{D}} \right| \\ & k=\sqrt{{{\mu }_{0}}{{\varepsilon }_{0}}}\omega {{\left( i\frac{{{J}_{C}}}{\left| {{J}_{D}} \right|} \right)}^{1/2}} \\ & =\sqrt{{{\mu }_{0}}{{\varepsilon }_{0}}}\omega {{\left( \frac{\sigma }{{{\varepsilon }_{0}}\omega } \right)}^{1/2}}{{\left( i \right)}^{1/2}} \\ & =\sqrt{{{\mu }_{0}}\sigma \omega }{{\left( \cos \frac{\pi }{2}+i\sin \frac{\pi }{2} \right)}^{1/2}} \\ & =\sqrt{{{\mu }_{0}}\sigma \omega }\left( \cos \frac{\pi }{4}+i\sin \frac{\pi }{4} \right) \\ & =\sqrt{{{\mu }_{0}}\sigma \omega }\left( \frac{1+i}{\sqrt{2}} \right) \\ & =\sqrt{\frac{{{\mu }_{0}}\sigma \omega }{2}}\left( 1+i \right) \\ & =\alpha +i\alpha \\ & E={{E}_{0}}{{e}^{i\left( \vec{k}\cdot \vec{r}-\omega t \right)}} \\ & ={{E}_{0}}{{e}^{i\left( kx-\omega t \right)}} \\ & ={{E}_{0}}{{e}^{i\left( \left( \alpha +i\alpha \right)x-\omega t \right)}} \\ & ={{E}_{0}}{{e}^{-\alpha x}}{{e}^{i\left( \alpha x-\omega t \right)}} \\ & =E_{0}^{'}{{e}^{i\left( \alpha x-\omega t \right)}} \\ & \Rightarrow E_{0}^{'}={{E}_{0}}{{e}^{-\alpha x}} \\ & \Rightarrow \frac{{{E}_{0}}}{e}={{E}_{0}}{{e}^{-\alpha \delta }} \\ & \Rightarrow {{e}^{-1}}={{e}^{-\alpha \delta }} \\ & \Rightarrow \delta =\frac{1}{\alpha } \\ & \Rightarrow \delta =\sqrt{\frac{2}{{{\mu }_{0}}\omega \sigma }} \\ \end{align}\]

Average Pyonting Vector

\[\begin{align} \left\langle S \right\rangle & =\left\langle \frac{\vec{E}\times \vec{B}}{{{\mu }_{0}}} \right\rangle \\ & =\left\langle \frac{EB\operatorname{Sin}90}{v\mu } \right\rangle \\ & =\frac{\left\langle EB \right\rangle }{v\mu } \\ & =\frac{\left\langle {{E}^{2}} \right\rangle }{v\mu } \\ & =\frac{\left\langle E_{0}^{2}{{\cos }^{2}}\left( kx-\omega t \right) \right\rangle }{v\mu } \\ \end{align}\]

Magnetic Vector Potential

Magnetic Vector Potential is defined as the vector field the curl of which gives the magnetic field intensity, mathematically , \( \vec{B}=\nabla\times \vec{A}\)

The Curl of it gives Magnetic field intensity
If its divergence is zero it will represent coulomb gauge condition
The relation between magnetic vector potential and electric scalar potential is :\[ \vec{A}=\dfrac{v}{c^2}\phi\]
The vector potential is not unique
Its relation with current density is:\[ \vec{A}(\vec{r}) = \frac{\mu_0}{4\pi} \int \frac{\vec{J}(\vec{r}')}{|\vec{r}-\vec{r}'|} \, d^3r' \]

Expression for Magnetic vector Potential

\[\begin{align} & d\vec{B}=\frac{{{\mu }_{0}}}{4\pi }\frac{id\vec{l}\times \vec{r}}{{{r}^{3}}} \\ \Rightarrow \vec{B}& =\int{\frac{{{\mu }_{0}}}{4\pi }\frac{id\vec{l}\times \vec{r}}{{{r}^{3}}}} \\ & =\frac{{{\mu }_{0}}}{4\pi }\int{\frac{id\vec{l}\times \vec{r}}{{{r}^{3}}}} \\ & =\frac{{{\mu }_{0}}}{4\pi }\int{id\vec{l}\times \left( \frac{{\vec{r}}}{{{r}^{3}}} \right)} \\ & =\frac{{{\mu }_{0}}}{4\pi }\int{id\vec{l}\times \left( \frac{{\hat{r}}}{{{r}^{2}}} \right)} \\ & =\frac{{{\mu }_{0}}}{4\pi }\int{id\vec{l}\times \left( -\nabla \left( \frac{1}{r} \right) \right)} \\ & =\frac{{{\mu }_{0}}}{4\pi }\int{\left( \nabla \left( \frac{1}{r} \right)\times id\vec{l} \right)} \\ & =\frac{{{\mu }_{0}}}{4\pi }\int{\left( \nabla \times \left( \frac{id\vec{l}}{r} \right) \right)} \\ & =\nabla \times \int{\frac{{{\mu }_{0}}}{4\pi }\left( \frac{id\vec{l}}{r} \right)} \\ & =\nabla \times \vec{A} \\ \Rightarrow \vec{A}&=\frac{{{\mu }_{0}}}{4\pi }\int{\frac{id\vec{l}}{r}} \end{align}\]

Magnetic Field Intensity due to a current carrying straight wire

\[\begin{align} d\vec{A} &=\frac{{{\mu }_{0}}}{4\pi }\frac{idy}{R} \\ & =\frac{{{\mu }_{0}}}{4\pi }\frac{idy}{\sqrt{{{r}^{2}}+{{y}^{2}}}} \\ \Rightarrow \vec{A}& =\frac{{{\mu }_{0}}}{4\pi }\int\limits_{-L/2}^{+L/2}{\frac{idy}{\sqrt{{{r}^{2}}+{{y}^{2}}}}} \\ & =\frac{{{\mu }_{0}}}{4\pi }\int\limits_{-L/2}^{+L/2}{\frac{idy}{\sqrt{{{r}^{2}}+{{y}^{2}}}}} \\ & =\frac{{{\mu }_{0}}i}{4\pi }\left[ \ln \left( y+\sqrt{{{r}^{2}}+{{y}^{2}}} \right) \right]_{-L/2}^{+L/2} \\ & =\frac{{{\mu }_{0}}i}{4\pi }\left( \ln \frac{\left( \frac{L}{2}+\sqrt{{{r}^{2}}+{{\frac{L}{4}}^{2}}} \right)}{\left( -\frac{L}{2}+\sqrt{{{r}^{2}}+{{\frac{L}{4}}^{2}}} \right)} \right) \\ & =\frac{{{\mu }_{0}}i}{4\pi }\left( \ln \frac{\left( L+\sqrt{4{{r}^{2}}+{{L}^{2}}} \right)}{\left( -L+\sqrt{4{{r}^{2}}+{{L}^{2}}} \right)} \right) \\ & =\frac{{{\mu }_{0}}i}{4\pi }\left( \ln \frac{\left( L+L\sqrt{1+\frac{4{{r}^{2}}}{{{L}^{2}}}} \right)}{\left( -L+L\sqrt{1+\frac{4{{r}^{2}}}{{{L}^{2}}}} \right)} \right) \\ & =\frac{{{\mu }_{0}}i}{4\pi }\left( \ln \frac{\left( L+L\left( 1+\frac{2{{r}^{2}}}{{{L}^{2}}} \right) \right)}{\left( -L+L\left( 1+\frac{2{{r}^{2}}}{{{L}^{2}}} \right) \right)} \right) \\ & =\frac{{{\mu }_{0}}i}{4\pi }\left( \ln \frac{\left( 2L+\frac{2{{r}^{2}}}{L} \right)}{\left( -L+L+\frac{2{{r}^{2}}}{L} \right)} \right) \\ & =\frac{{{\mu }_{0}}i}{4\pi }\left( \ln \frac{\left( 2L+\frac{2{{r}^{2}}}{L} \right)}{\left( \frac{2{{r}^{2}}}{L} \right)} \right) \\ & =\frac{{{\mu }_{0}}i}{4\pi }\left( \ln \frac{\left( 2L \right)}{\left( \frac{2{{r}^{2}}}{L} \right)} \right) \\ & =\frac{{{\mu }_{0}}i}{4\pi }\ln \left( \frac{2{{L}^{2}}}{2{{r}^{2}}} \right) \\ & =2\times \frac{{{\mu }_{0}}i}{4\pi }\ln \left( \frac{L}{r} \right) \\ & =\frac{{{\mu }_{0}}i}{2\pi }\ln \left( \frac{L}{r} \right)\hat{j} \\ \Rightarrow \vec{B} &=\nabla \times \vec{A} \\ & =\left( \begin{matrix} {\hat{r}} & {\hat{\theta }} & {\hat{z}} \\ \frac{\partial }{\partial r} & \frac{1}{r}\frac{\partial }{\partial r} & \frac{\partial }{\partial z} \\ {{A}_{r}} & {{A}_{\theta }} & {{A}_{z}} \\ \end{matrix} \right) \\ & =\left( \begin{matrix} {\hat{r}} & {\hat{\theta }} & {\hat{z}} \\ \frac{\partial }{\partial r} & \frac{1}{r}\frac{\partial }{\partial r} & \frac{\partial }{\partial z} \\ 0 & \frac{{{\mu }_{0}}i}{2\pi }\ln \left( \frac{L}{r} \right) & 0 \\ \end{matrix} \right) \\ & =\hat{z}\left( \frac{\partial }{\partial r}\frac{{{\mu }_{0}}i}{2\pi }\ln \left( \frac{L}{r} \right)-\frac{1}{r}\frac{\partial }{\partial r}\left( 0 \right) \right) \\ & \because \vec{B}=\left( -\frac{{{\mu }_{0}}i}{2\pi r} \right)\hat{z} \\ \end{align}\]

Magnetic Field Intensity due to a current carrying loop of radius r

\[\begin{align} A &=\int{d{{A}_{y}}} \\ & =\int\limits_{0}^{2\pi }{\frac{{{\mu }_{0}}}{4\pi }}\frac{idl\cos \phi }{r'} \\ & =\frac{{{\mu }_{0}}i}{4\pi }\int\limits_{0}^{2\pi }{\frac{dl\cos \phi }{r'}} \\ & =\frac{{{\mu }_{0}}i}{4\pi }\int\limits_{0}^{2\pi }{\frac{ad\phi \cos \phi }{\sqrt{{{r}^{2}}+{{a}^{2}}-2ra\cos \alpha }}} \\ & =\frac{{{\mu }_{0}}i}{4\pi }\int\limits_{0}^{2\pi }{\frac{ad\phi \cos \phi }{{{r}^{2}}\sqrt{1+{{\frac{a}{{{r}^{2}}}}^{2}}-2\frac{a}{r}\cos \alpha }}} \\ & =\frac{{{\mu }_{0}}i}{4\pi }\int\limits_{0}^{2\pi }{\frac{ad\phi \cos \phi }{r\sqrt{1-2\frac{a}{r}\cos \alpha }}} \\ & =\frac{{{\mu }_{0}}i}{4\pi }\int\limits_{0}^{2\pi }{\frac{ad\phi \cos \phi }{r\left( 1-\frac{1}{2}\times \frac{2a}{r}\cos \alpha \right)}} \\ & =\frac{{{\mu }_{0}}i}{4\pi }\int\limits_{0}^{2\pi }{\frac{ad\phi \cos \phi }{r\left( 1-\frac{a}{r}\cos \alpha \right)}} \\ & =\frac{{{\mu }_{0}}i}{4\pi }\int\limits_{0}^{2\pi }{\frac{ad\phi \cos \phi }{r}}{{\left( 1-\frac{a}{r}\cos \alpha \right)}^{-1}} \\ & =\frac{{{\mu }_{0}}i}{4\pi }\int\limits_{0}^{2\pi }{\frac{ad\phi \cos \phi }{r}}\left( 1+\frac{a}{r}\cos \alpha \right) \\ & =\frac{{{\mu }_{0}}i}{4\pi }\int\limits_{0}^{2\pi }{\frac{ad\phi \cos \phi }{r}}\left( 1+\frac{a}{r}\frac{x\cos \phi }{r} \right) \\ & =\frac{{{\mu }_{0}}i}{4\pi }\int\limits_{0}^{2\pi }{\frac{ad\phi \cos \phi }{r}}\left( 1+\frac{ax\cos \phi }{{{r}^{2}}} \right) \\ & =\frac{{{\mu }_{0}}i}{4\pi }\int\limits_{0}^{2\pi }{\left( \frac{a\cos \phi }{r}+\frac{{{a}^{2}}x{{\cos }^{2}}\phi }{{{r}^{3}}} \right)}\,d\phi \\ & =\frac{{{\mu }_{0}}i}{4\pi }\left( 0+\frac{{{a}^{2}}x}{{{r}^{3}}}\pi \right) \\ & =\frac{{{\mu }_{0}}}{4\pi }\left( \frac{i\pi {{a}^{2}}x}{{{r}^{3}}} \right) \\ & =\frac{{{\mu }_{0}}}{4\pi }\left( \frac{mx}{{{r}^{3}}} \right) \\ & =\frac{{{\mu }_{0}}}{4\pi }\left( \frac{mr\sin \theta }{{{r}^{3}}} \right) \\ & =\frac{{{\mu }_{0}}}{4\pi }\left( \frac{\vec{m}\times \vec{r}}{{{r}^{3}}} \right) \\ \vec{B}& =\nabla \times \vec{A} \\ & =\nabla \times \left( \frac{{{\mu }_{0}}}{4\pi }\left( \frac{\vec{m}\times \vec{r}}{{{r}^{3}}} \right) \right) \\ & =\frac{{{\mu }_{0}}}{4\pi }\nabla \times \left( \vec{m}\times \frac{{\vec{r}}}{{{r}^{3}}} \right) \\ & =\frac{{{\mu }_{0}}}{4\pi }\left( \vec{m}\left( \nabla \centerdot \frac{{\vec{r}}}{{{r}^{3}}} \right)-\frac{{\vec{r}}}{{{r}^{3}}}\left( \nabla \centerdot \vec{m} \right)+\left( \frac{{\vec{r}}}{{{r}^{3}}}\centerdot \nabla \right)\vec{m}-\left( \vec{m}\centerdot \nabla \right)\frac{{\vec{r}}}{{{r}^{3}}} \right) \\ & =\frac{{{\mu }_{0}}}{4\pi }\left( \vec{m}\left( \nabla \centerdot \frac{{\hat{r}}}{{{r}^{2}}} \right)-\frac{{\vec{r}}}{{{r}^{3}}}\left( \nabla \centerdot \vec{m} \right)+\left( \frac{{\vec{r}}}{{{r}^{3}}}\centerdot \nabla \right)\vec{m}-\left( \vec{m}\centerdot \nabla \right)\frac{{\vec{r}}}{{{r}^{3}}} \right) \\ & =\frac{{{\mu }_{0}}}{4\pi }\left( \vec{m}\left( 0 \right)-\frac{{\vec{r}}}{{{r}^{3}}}\left( 0 \right)+\left( \frac{{\vec{r}}}{{{r}^{3}}}\centerdot \nabla \right)\vec{m}-\left( \vec{m}\centerdot \nabla \right)\frac{{\vec{r}}}{{{r}^{3}}} \right) \\ & =\frac{{{\mu }_{0}}}{4\pi }\left( \left( \frac{{\vec{r}}}{{{r}^{3}}}\centerdot \nabla \right)\vec{m}-\left( \vec{m}\centerdot \nabla \right)\frac{{\vec{r}}}{{{r}^{3}}} \right) \\ & =\frac{{{\mu }_{0}}}{4\pi }\left( \left( \frac{x\partial }{{{r}^{3}}\partial x}+\frac{y\partial }{{{r}^{3}}\partial x}+\frac{z\partial }{{{r}^{3}}\partial z} \right)\vec{m}-\left( \vec{m}\centerdot \nabla \right)\frac{{\vec{r}}}{{{r}^{3}}} \right) \\ & =\frac{{{\mu }_{0}}}{4\pi }\left( 0-\left( \vec{m}\centerdot \nabla \right)\frac{{\vec{r}}}{{{r}^{3}}} \right) \\ & =-\frac{{{\mu }_{0}}}{4\pi }\left( \vec{m}\centerdot \nabla \right)\frac{{\vec{r}}}{{{r}^{3}}} \\ & =-\frac{{{\mu }_{0}}}{4\pi }\left( {{m}_{x}}\frac{\partial }{\partial x}+{{m}_{y}}\frac{\partial }{\partial y}+{{m}_{z}}\frac{\partial }{\partial z} \right)\frac{{\vec{r}}}{{{r}^{3}}} \\ & =-\frac{{{\mu }_{0}}}{4\pi }\left( {{m}_{x}}\frac{\partial }{\partial x}\left( \frac{{\vec{r}}}{{{r}^{3}}} \right)+{{m}_{y}}\frac{\partial }{\partial y}\left( \frac{{\vec{r}}}{{{r}^{3}}} \right)+{{m}_{z}}\frac{\partial }{\partial z}\left( \frac{{\vec{r}}}{{{r}^{3}}} \right) \right) \\ & =-\frac{{{\mu }_{0}}}{4\pi }\left( {{m}_{x}}\left( \frac{\hat{i}{{r}^{3}}-3rx\vec{r}}{{{r}^{6}}} \right)+{{m}_{y}}\left( \frac{\hat{j}{{r}^{2}}-3ry\vec{r}}{{{r}^{6}}} \right)+{{m}_{z}}\left( \frac{\hat{k}{{r}^{2}}-3rz\vec{r}}{{{r}^{6}}} \right) \right) \\ & =\frac{{{\mu }_{0}}}{4\pi }\left( \left( \frac{3{{m}_{x}}x\vec{r}}{{{r}^{5}}}+\frac{3{{m}_{y}}y\vec{r}}{{{r}^{5}}}+\frac{3{{m}_{z}}z\vec{r}}{{{r}^{5}}} \right)-\frac{{\vec{m}}}{{{r}^{3}}} \right) \\ & =\frac{{{\mu }_{0}}}{4\pi }\left( \left( \frac{3\left( \vec{m}\cdot \vec{r} \right)\vec{r}}{{{r}^{5}}} \right)-\frac{{\vec{m}}}{{{r}^{3}}} \right) \\ \end{align}\] \tag{5}

Linerd wiechert Potentials

The Liénard–Wiechert potentials are the exact solutions of Maxwell’s equations that describe the scalar potential and the vector potential produced by a point charge moving arbitrarily in space with velocity They are extremely important in electrodynamics because they give the correct electric and magnetic fields due to an accelerated moving charge, including radiation.

They include both Coulomb-like near fields (falling off as \( 1/R^2 \) and radiation fields (falling off as \( 1/R \)
When the charge is at rest, they reduce to the familiar Coulomb potential
When the charge accelerates, they naturally give rise to electromagnetic radiation.

Applications

Radiation from accelerating charges (synchrotron radiation, bremsstrahlung, dipole radiation
Foundation for understanding electromagnetic waves emitted by antennas.
Starting point for deriving the Lienard formula for radiated power

\[ \phi(\vec{r},t) = \frac{1}{4\pi\epsilon_0} \frac{q}{\left(1 - \dfrac{\vec{v}(t_r) \cdot \hat{R}}{c}\right) R} \] \[ \vec{A}(\vec{r},t) = \frac{\mu_0}{4\pi} \frac{q \, \vec{v}(t_r)}{\left(1 - \dfrac{\vec{v}(t_r) \cdot \hat{R}}{c}\right) R} \]

\[ t_r = t - \frac{R}{c}, \qquad \vec{R} = \vec{r} - \vec{r}_q(t_r), \qquad R = |\vec{R}|, \quad \hat{R} = \frac{\vec{R}}{R} \] \[ \vec{E}(\vec{r},t) = \frac{q}{4\pi \epsilon_0} \left[ \frac{(1 - \beta^2)(\hat{R} - \vec{\beta})}{\left(1 - \hat{R}\cdot\vec{\beta}\right)^3 R^2} + \frac{\hat{R} \times \left( (\hat{R} - \vec{\beta}) \times \dot{\vec{\beta}} \right)}{c \left(1 - \hat{R}\cdot\vec{\beta}\right)^3 R} \right]_{t_r} \] \[ \vec{B}(\vec{r},t) = \hat{R} \times \vec{E}(\vec{r},t) \] \[ \vec{\beta} = \frac{\vec{v}(t_r)}{c}, \qquad \dot{\vec{\beta}} = \frac{\vec{a}(t_r)}{c}, \qquad \hat{R} = \frac{\vec{r} - \vec{r}_q(t_r)}{|\vec{r} - \vec{r}_q(t_r)|} \] \[ \vec{E}(\vec{r},t) = \frac{q}{4\pi \epsilon_0} \left[ \frac{(1 - \beta^2)(\hat{R} - \vec{\beta})}{(1 - \hat{R}\cdot\vec{\beta})^3 R^2} + \frac{\hat{R} \times ((\hat{R} - \vec{\beta}) \times \dot{\vec{\beta}})}{c (1 - \hat{R}\cdot\vec{\beta})^3 R} \right]_{t_r} \] \[ \vec{B}(\vec{r},t) = \hat{R} \times \vec{E}(\vec{r},t) \]

Derivation of E and B from Lienerd Wiechert Potentials

\[\begin{align*} & \phi =\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{qc}{\left( Rc-\vec{R}\cdot \vec{v} \right)} \\ \vec{A}& =\frac{{\vec{v}}}{{{c}^{2}}}\phi \\ & =\frac{1}{4\pi {{\varepsilon }_{0}}{{c}^{2}}}\frac{q\vec{v}}{\left( Rc-\vec{R}\cdot \vec{v} \right)} \\ & =\frac{{{\mu }_{0}}{{\varepsilon }_{0}}}{4\pi {{\varepsilon }_{0}}}\frac{q\vec{v}}{\left( Rc-\vec{R}\cdot \vec{v} \right)} \\ & =\frac{{{\mu }_{0}}}{4\pi }\frac{q\vec{v}}{\left( Rc-\vec{R}\cdot \vec{v} \right)} \\ & \vec{E}=-\nabla \phi -\frac{\partial \vec{A}}{\partial t} \\ \Rightarrow \nabla \phi& =\nabla \left( \frac{1}{4\pi {{\varepsilon }_{0}}}\frac{qc}{\left( Rc-\vec{R}\cdot \vec{v} \right)} \right) \\ & =\frac{qc}{4\pi {{\varepsilon }_{0}}}\nabla {{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{-1}} \\ & =\frac{qc}{4\pi {{\varepsilon }_{0}}}\frac{1}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}}\nabla \left( \vec{R}\cdot \vec{v}-Rc \right) \\ & =\frac{qc}{4\pi {{\varepsilon }_{0}}}\frac{1}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}}\left( \nabla \left( \vec{R}\cdot \vec{v} \right)-c\nabla R \right) \\ & =\frac{qc}{4\pi {{\varepsilon }_{0}}}\frac{1}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}}\left( \left( \vec{R} \cdot\nabla \right)\vec{v}+\left( \vec{v} \cdot \nabla \right)\vec{R}+\vec{R}\times \left( \nabla \times \vec{v} \right)+\vec{v}\times \left( \nabla \times \vec{R} \right)-c\nabla R \right) \\ \Rightarrow \nabla \phi&=\frac{qc}{4\pi {{\varepsilon }_{0}}}\frac{1}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}}\left( \alpha +\beta +\gamma +\delta -c\nabla R \right)\tag{5} \\ \end{align*}\]

Let us evaluate \( \alpha \)

\[\begin{align*} \vec{\alpha } &=\left( \vec{R}\cdot \nabla \right)\vec{v} \\ & =\left( {{R}_{x}}\frac{\partial }{\partial x}+{{R}_{y}}\frac{\partial }{\partial y}+{{R}_{z}}\frac{\partial }{\partial z} \right)\vec{v} \\ & ={{R}_{x}}\frac{\partial \vec{v}}{\partial x}+{{R}_{y}}\frac{\partial \vec{v}}{\partial y}+{{R}_{z}}\frac{\partial \vec{v}}{\partial z} \\ & ={{R}_{x}}\frac{\partial t'}{\partial x}\frac{\partial \vec{v}}{\partial t'}+{{R}_{y}}\frac{\partial t'}{\partial y}\frac{\partial \vec{v}}{\partial t'}+{{R}_{z}}\frac{\partial t'}{\partial z}\frac{\partial \vec{v}}{\partial t'} \\ & =\left( {{R}_{x}}\frac{\partial t'}{\partial x}+{{R}_{y}}\frac{\partial t'}{\partial y}+{{R}_{z}}\frac{\partial t'}{\partial z} \right)\frac{\partial \vec{v}}{\partial t'} \\ & =\left( \vec{R}\cdot \nabla t' \right)\frac{\partial \vec{v}}{\partial t'} \\ & \Rightarrow \vec{\alpha }=\left( \vec{R}\cdot \nabla t' \right)\vec{a}\tag{5.1} \\ \end{align*}\]

Let us evaluate \( \beta \)

\[\begin{align} \vec{\beta } &=\left( \vec{v}\cdot \nabla \, \right)\vec{R} \\ & =\left( {{v}_{x}}\frac{\partial }{\partial x}+{{v}_{y}}\frac{\partial }{\partial y}+{{v}_{z}}\frac{\partial }{\partial z} \right)\vec{R} \\ & =\left( {{v}_{x}}\frac{\partial }{\partial x}+{{v}_{y}}\frac{\partial }{\partial y}+{{v}_{z}}\frac{\partial }{\partial z} \right)\left( \vec{r}-\vec{r}' \right) \\ & =\left( {{v}_{x}}\frac{\partial \vec{r}}{\partial x}+{{v}_{y}}\frac{\partial \vec{r}}{\partial y}+{{v}_{z}}\frac{\partial \vec{r}}{\partial z} \right)-\left( {{v}_{x}}\frac{\partial \vec{r}'}{\partial x}+{{v}_{y}}\frac{\partial \vec{r}'}{\partial y}+{{v}_{z}}\frac{\partial \vec{r}'}{\partial z} \right) \\ & =\left( {{v}_{x}}i+{{v}_{y}}j+{{v}_{z}}z \right)-\left( {{v}_{x}}\frac{\partial \vec{r}'}{\partial t'}\frac{\partial t'}{\partial x}+{{v}_{y}}\frac{\partial \vec{r}'}{\partial t'}\frac{\partial t'}{\partial y}+{{v}_{z}}\frac{\partial \vec{r}'}{\partial t'}\frac{\partial t'}{\partial z} \right) \\ & =\vec{v}-\left( {{v}_{x}}\frac{\partial t'}{\partial x}+{{v}_{y}}\frac{\partial t'}{\partial y}+{{v}_{z}}\frac{\partial t'}{\partial z} \right)\frac{\partial \vec{r}'}{\partial t'} \\ & =\vec{v}-\left( \vec{v}\cdot \nabla t' \right)\vec{v} \\ & =\vec{v}\left( 1-\left( \vec{v}\cdot \nabla t' \right) \right) \\ \end{align}\]

Let us find \( \gamma \)

\[\begin{align} & \vec{\gamma }=\vec{R}\times \left( \nabla \times \vec{v} \right) \\ & =\vec{R}\times \left| \begin{matrix} i & j & k \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ {{v}_{x}} & {{v}_{y}} & {{v}_{z}} \\ \end{matrix} \right| \\ & =\vec{R}\times \left( \left( \frac{\partial {{v}_{z}}}{\partial y}-\frac{\partial {{v}_{y}}}{\partial z} \right)\hat{i}+\left( \frac{\partial {{v}_{x}}}{\partial z}-\frac{\partial {{v}_{z}}}{\partial x} \right)\hat{j}+\left( \frac{\partial {{v}_{y}}}{\partial x}-\frac{\partial {{v}_{x}}}{\partial y} \right)\hat{k} \right) \\ & =\vec{R}\times \left( \left( \frac{\partial {{v}_{z}}}{\partial t'}\frac{\partial t'}{\partial y}-\frac{\partial {{v}_{y}}}{\partial t'}\frac{\partial t'}{\partial z} \right)\hat{i}+\left( \frac{\partial {{v}_{x}}}{\partial t'}\frac{\partial t'}{\partial z}-\frac{\partial {{v}_{z}}}{\partial t'}\frac{\partial t'}{\partial z} \right)\hat{j}+\left( \frac{\partial {{v}_{y}}}{\partial t'}\frac{\partial t'}{\partial t'}-\frac{\partial {{v}_{x}}}{\partial t'}\frac{\partial t'}{\partial y} \right)\hat{k} \right) \\ & =-\vec{R}\times \left( \left( \frac{\partial {{v}_{y}}}{\partial t'}\frac{\partial t'}{\partial z}-\frac{\partial {{v}_{z}}}{\partial t'}\frac{\partial t'}{\partial y} \right)\hat{i}+\left( \frac{\partial {{v}_{z}}}{\partial t'}\frac{\partial t'}{\partial z}-\frac{\partial {{v}_{x}}}{\partial t'}\frac{\partial t'}{\partial x} \right)\hat{j}+\left( \frac{\partial {{v}_{x}}}{\partial t'}\frac{\partial t'}{\partial y}-\frac{\partial {{v}_{y}}}{\partial t'}\frac{\partial t'}{\partial y} \right)\hat{k} \right) \\ & =-\vec{R}\times \left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ \frac{\partial {{v}_{x}}}{\partial t'} & \frac{\partial {{v}_{y}}}{\partial t'} & \frac{\partial {{v}_{z}}}{\partial t'} \\ \frac{\partial t'}{\partial x} & \frac{\partial t'}{\partial y} & \frac{\partial t'}{\partial z} \\ \end{matrix} \right| \\ & =-\vec{R}\times \left( \left( \frac{\partial {{v}_{x}}}{\partial t'}\hat{i}+\frac{\partial {{v}_{y}}}{\partial t'}\hat{j}+\frac{\partial {{v}_{z}}}{\partial t'}\hat{k} \right)\times \left( \frac{\partial t'}{\partial x}\hat{i}+\frac{\partial t'}{\partial y}\hat{j}+\frac{\partial t'}{\partial z}\hat{k} \right) \right) \\ & =-\vec{R}\times \left( \left( {{a}_{x}}\hat{i}+{{a}_{y}}\hat{j}+{{a}_{z}}\hat{k} \right)\times \left( \nabla t' \right) \right) \\ & =-\vec{R}\times \left( \vec{a}\times \nabla t' \right) \\ & \Rightarrow \vec{\gamma }=-\vec{R}\times \left( \vec{a}\times \nabla t' \right) \\ \end{align}\]

Let us find \( \vec{\delta } \)

\[\begin{align} & \vec{\delta }=\vec{v}\times \left( \nabla \times \vec{R} \right) \\ & =\vec{v}\times \left( \nabla \times \left( \vec{r}-\vec{r}' \right) \right) \\ & =\vec{v}\times \left( \nabla \times \vec{r}-\nabla \times \vec{r}' \right) \\ & =\vec{v}\times \left( \nabla \times \vec{r} \right)-\vec{v}\times \left( \nabla \times \vec{r}' \right) \\ & =-\vec{v}\times \left( \nabla \times \vec{r}' \right) \\ & =-\vec{v}\times \left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ x' & y' & z' \\ \end{matrix} \right| \\ & =-\vec{v}\times \left( \left( \frac{\partial z'}{\partial y}-\frac{\partial y'}{\partial y} \right)\hat{i}+\left( \frac{\partial x'}{\partial z}-\frac{\partial z'}{\partial x} \right)\hat{j}+\left( \frac{\partial y'}{\partial x}-\frac{\partial x'}{\partial y} \right)\hat{k} \right) \\ & =-\vec{v}\times \left( \left( \frac{\partial z'}{\partial t'}\frac{\partial t'}{\partial y}-\frac{\partial y'}{\partial t'}\frac{\partial t'}{\partial y} \right)\hat{i}+\left( \frac{\partial x'}{\partial t'}\frac{\partial t'}{\partial z}-\frac{\partial z'}{\partial t'}\frac{\partial t'}{\partial x} \right)\hat{j}+\left( \frac{\partial y'}{\partial t'}\frac{\partial t'}{\partial x}-\frac{\partial x'}{\partial t'}\frac{\partial t'}{\partial y} \right)\hat{k} \right) \\ & =\vec{v}\times \left( \left( \frac{\partial y'}{\partial t'}\frac{\partial t'}{\partial y}-\frac{\partial z'}{\partial t'}\frac{\partial t'}{\partial y} \right)\hat{i}+\left( \frac{\partial z'}{\partial t'}\frac{\partial t'}{\partial x}-\frac{\partial x'}{\partial t'}\frac{\partial t'}{\partial z} \right)\hat{j}+\left( \frac{\partial x'}{\partial t'}\frac{\partial t'}{\partial y}-\frac{\partial y'}{\partial t'}\frac{\partial t'}{\partial x} \right)\hat{k} \right) \\ & =\vec{v}\times \left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ \frac{\partial x'}{\partial t'} & \frac{\partial y'}{\partial t'} & \frac{\partial z'}{\partial t'} \\ \frac{\partial t'}{\partial x} & \frac{\partial t'}{\partial y} & \frac{\partial t'}{\partial z} \\ \end{matrix} \right| \\ & =\vec{v}\times \left( \left( \frac{\partial x'}{\partial t'}\hat{i}+\frac{\partial y'}{\partial t'}\hat{j}+\frac{\partial z'}{\partial t'}\hat{k} \right)\times \left( \frac{\partial t'}{\partial x}\hat{i}+\frac{\partial t'}{\partial y}\hat{j}+\frac{\partial t'}{\partial z}\hat{k} \right) \right) \\ & =\vec{v}\times \left( \left( \frac{\partial \vec{r}'}{\partial t'} \right)\times \left( \frac{\partial }{\partial x}\hat{i}+\frac{\partial }{\partial y}\hat{j}+\frac{\partial }{\partial z}\hat{k} \right)\,t' \right) \\ & =\vec{v}\times \left( \vec{v}\times \nabla \,t' \right) \\ & \Rightarrow \vec{\delta }=\vec{v}\times \left( \vec{v}\times \nabla \,t' \right) \\ \end{align}\]

Let us simplify \( c\nabla R \)

\[\begin{align} & c\nabla R=c\nabla \left( ct-c{{t}_{r}} \right) \\ & ={{c}^{2}}\nabla t-{{c}^{2}}\nabla {{t}_{r}}\because \left( {{t}_{r}}=t-t'=t-\frac{R}{c} \right) \\ & =-{{c}^{2}}\nabla {{t}_{r}} \\ & \Rightarrow \nabla R=-c\nabla {{t}_{r}} \\ & \Rightarrow \nabla \sqrt{\vec{R}\cdot \vec{R}}=-c\nabla {{t}_{r}} \\ & \Rightarrow \frac{\nabla \left( \vec{R}\cdot \vec{R} \right)}{2\sqrt{\vec{R}\cdot \vec{R}}}=-c\nabla {{t}_{r}} \\ & \Rightarrow \frac{\left( \nabla \cdot \vec{R} \right)\vec{R}+\left( \nabla \cdot \vec{R} \right)\vec{R}+\vec{R}\times \left( \nabla \times \vec{R} \right)+\vec{R}\times \left( \nabla \times \vec{R} \right)}{2\sqrt{\vec{R}\cdot \vec{R}}}=-c\nabla {{t}_{r}} \\ & \Rightarrow \frac{2\left( \nabla \cdot \vec{R} \right)\vec{R}+2\vec{R}\times \left( \nabla \times \vec{R} \right)}{2\sqrt{\vec{R}\cdot \vec{R}}}=-c\nabla {{t}_{r}} \\ & \Rightarrow \frac{\vec{R}-\vec{v}\left( \vec{R}\cdot \nabla {{t}_{r}} \right)+\vec{R}\times \left( \vec{v}\times \nabla {{t}_{r}} \right)}{\sqrt{\vec{R}\cdot \vec{R}}}=-c\nabla {{t}_{r}} \\ & \Rightarrow \frac{\vec{R}-\vec{v}\left( \vec{R}\cdot \nabla {{t}_{r}} \right)+\vec{R}\times \left( \vec{v}\times \nabla {{t}_{r}} \right)}{R}=-c\nabla {{t}_{r}} \\ & \Rightarrow \vec{R}-\vec{v}\left( \vec{R}\cdot \nabla {{t}_{r}} \right)+\vec{R}\times \left( \vec{v}\times \nabla {{t}_{r}} \right)=-cR\nabla {{t}_{r}} \\ & \Rightarrow \vec{R}-\vec{v}\left( \vec{R}\cdot \nabla {{t}_{r}} \right)+\left( \vec{R}\cdot \nabla {{t}_{r}} \right)\vec{v}-\left( \vec{R}\cdot \vec{v} \right)\nabla {{t}_{r}}=-cR\nabla {{t}_{r}} \\ & \Rightarrow \vec{R}-\left( \vec{R}\cdot \vec{v} \right)\nabla {{t}_{r}}=-cR\nabla {{t}_{r}} \\ & \Rightarrow \nabla {{t}_{r}}\left( Rc-\left( \vec{R}\cdot \vec{v} \right) \right)=-\vec{R} \\ & \Rightarrow \nabla {{t}_{r}}=\frac{-\vec{R}}{Rc-\left( \vec{R}\cdot \vec{v} \right)} \\ & c\nabla R=-{{c}^{2}}\nabla {{t}_{r}} \\ & \Rightarrow c\nabla R=\frac{{{c}^{2}}R}{Rc-\vec{R}\cdot \vec{v}} \\ \end{align}\] \[\begin{align} & c\nabla R=-{{c}^{2}}\nabla {{t}_{r}} \\ & \Rightarrow c\nabla R=\frac{{{c}^{2}}R}{Rc-\vec{R}\cdot \vec{v}} \\ & \nabla \phi =\frac{qc}{4\pi {{\varepsilon }_{0}}{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}}\left( \left( \vec{R}\cdot \nabla t' \right)\vec{a}+\vec{v}\left( 1-\left( \vec{v}\cdot \nabla t' \right) \right)-\vec{R}\times \left( \vec{a}\times \nabla t' \right)+\vec{v}\times \left( \vec{v}\times \nabla t' \right)-c\nabla R \right) \\ & =\frac{qc}{4\pi {{\varepsilon }_{0}}{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}}\left( \left( \vec{R}\cdot \nabla t' \right)\vec{a}+\vec{v}\left( 1-\left( \vec{v}\cdot \nabla t' \right) \right)-\left( \vec{R}\cdot \nabla t' \right)\vec{a}+\left( \vec{R}\cdot \vec{a} \right)\nabla t'+\left( \vec{v}\cdot \nabla t' \right)\vec{v}-{{v}^{2}}\nabla t'-c\nabla R \right) \\ & =\frac{qc}{4\pi {{\varepsilon }_{0}}{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}}\left( \vec{v}+\left( \vec{R}\cdot \vec{a} \right)\nabla t'-{{v}^{2}}\nabla t'-{{c}^{2}}\nabla t' \right) \\ & =\frac{qc}{4\pi {{\varepsilon }_{0}}{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}}\left( \vec{v}+\left( \left( \vec{R}\cdot \vec{a} \right)-{{v}^{2}}-{{c}^{2}} \right)\nabla t' \right) \\ & =\frac{qc}{4\pi {{\varepsilon }_{0}}{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}}\left( \vec{v}+\left( \left( \vec{R}\cdot \vec{a} \right)-{{v}^{2}}-{{c}^{2}} \right)\nabla \left( t-{{t}_{r}} \right) \right) \\ & =-\frac{qc}{4\pi {{\varepsilon }_{0}}{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}}\left( \vec{v}+\left( \left( \vec{R}\cdot \vec{a} \right)-{{v}^{2}}-{{c}^{2}} \right)\nabla {{t}_{r}} \right) \\ & =-\frac{qc}{4\pi {{\varepsilon }_{0}}{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}}\left( \vec{v}+\left( \left( \vec{R}\cdot \vec{a} \right)-{{v}^{2}}-{{c}^{2}} \right)\frac{-\vec{R}}{Rc-\vec{R}\cdot \vec{v}} \right) \\ & \Rightarrow \nabla \phi =\frac{qc}{4\pi {{\varepsilon }_{0}}{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left[ \left( Rc-\vec{R}\cdot \vec{v} \right)\vec{v}-\left( {{c}^{2}}+{{v}^{2}}-\vec{R}\cdot \vec{a} \right)\vec{R} \right] \\ \end{align}\]

Evaluation of \( \dfrac{\partial t'}{\partial t}\)

\[\begin{align} & {{t}_{r}}=t-t'=t-\frac{R}{c} \\ & \Rightarrow R=c\left( t-t' \right) \\ & \Rightarrow \vec{R}\cdot \vec{R}={{c}^{2}}{{\left( t-t' \right)}^{2}} \\ & \Rightarrow \frac{\partial }{\partial t}\left( \vec{R}\cdot \vec{R} \right)={{c}^{2}}\frac{\partial }{\partial t}{{\left( t-t' \right)}^{2}} \\ & \Rightarrow 2\left( \frac{\partial \vec{R}}{\partial t}\cdot \vec{R} \right)={{c}^{2}}\frac{\partial }{\partial t}{{\left( t-t' \right)}^{2}} \\ & \Rightarrow 2\left( \frac{\partial }{\partial t}\left( \vec{r}-\vec{r}' \right)\cdot \vec{R} \right)=2{{c}^{2}}\left( t-t' \right)\left( \frac{\partial t}{\partial t}-\frac{\partial t'}{\partial t} \right) \\ & \Rightarrow \left( \left( \frac{\partial \vec{r}}{\partial t}-\frac{\partial \vec{r}'}{\partial t'}\frac{\partial t'}{\partial t} \right)\cdot \vec{R} \right)=cc\left( t-t' \right)\left( 1-\frac{\partial t'}{\partial t} \right) \\ & \Rightarrow \left( \left( 0-\vec{v}\frac{\partial t'}{\partial t} \right)\cdot \vec{R} \right)=cR\left( 1-\frac{\partial t'}{\partial t} \right) \\ & \Rightarrow -\vec{v}\cdot \vec{R}\frac{\partial t'}{\partial t}=cR-cR\frac{\partial t'}{\partial t} \\ & \Rightarrow \frac{\partial t'}{\partial t}\left( Rc-\vec{v}\cdot \vec{R} \right)=Rc \\ & \Rightarrow \frac{\partial t'}{\partial t}=\frac{Rc}{Rc-\vec{v}\cdot \vec{R}} \\ \frac{\partial \vec{R}}{\partial t}&=\frac{Rc\vec{v}}{Rc-\vec{v}\cdot \vec{R}} \end{align}\]

Evaluation of \( \dfrac{\partial \vec{A}}{\partial t}\)

\[\begin{align} \frac{\partial \vec{A}}{\partial t} &=\frac{\partial }{\partial t}\left( \frac{{{\mu }_{0}}}{4\pi }\frac{qvc}{\left( Rc-\vec{R}\cdot \vec{v} \right)} \right) \\ & =\frac{{{\mu }_{0}}qc}{4\pi }\frac{\partial }{\partial t}\left( \frac{v}{\left( Rc-\vec{R}\cdot \vec{v} \right)} \right) \\ & =\frac{{{\mu }_{0}}qc}{4\pi }\left( \frac{\frac{\partial v}{\partial t}\left( Rc-\vec{R}\cdot \vec{v} \right)-v\frac{\partial }{\partial t}\left( Rc-\vec{R}\cdot \vec{v} \right)}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}} \right) \\ & =\frac{{{\mu }_{0}}qc}{4\pi }\left( \frac{\frac{\partial t'}{\partial t}\frac{\partial v}{\partial t'}\left( Rc-\vec{R}\cdot \vec{v} \right)-v\left( c\frac{\partial R}{\partial t}-\frac{\partial }{\partial t}\left( \vec{R}\cdot \vec{v} \right) \right)}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}} \right) \\ & =\frac{{{\mu }_{0}}qc}{4\pi }\left( \frac{\frac{Rc}{Rc-\vec{R}\cdot \vec{v}}a\left( Rc-\vec{R}\cdot \vec{v} \right)-v\left( c\frac{\partial R}{\partial t}-\frac{\partial \vec{R}}{\partial t}\cdot \vec{v}-\vec{R}\cdot \frac{\partial \vec{v}}{\partial t} \right)}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}} \right) \\ & =\frac{{{\mu }_{0}}qc}{4\pi }\left( \frac{Rca-v\left( {{c}^{2}}\frac{\partial \left( t-t' \right)}{\partial t}-\frac{\partial \vec{R}}{\partial t}\cdot \vec{v}-\vec{R}\cdot \frac{\partial \vec{v}}{\partial t} \right)}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}} \right) \\ & =\frac{{{\mu }_{0}}qc}{4\pi }\left( \frac{Rca-v\left( {{c}^{2}}\frac{\partial \left( t-t' \right)}{\partial t}-\frac{\partial \vec{R}}{\partial t}\cdot \vec{v}-\vec{R}\cdot \frac{Rc}{Rc-\vec{R}\cdot \vec{v}}\vec{a} \right)}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}} \right) \\ & =\frac{{{\mu }_{0}}qc}{4\pi }\left( \frac{Rca-v\left( {{c}^{2}}\left( 1-\frac{\partial t'}{\partial t} \right)-\left( -\vec{v} \right)\cdot \vec{v}-\vec{R}\cdot \frac{Rc}{Rc-\vec{R}\cdot \vec{v}}\vec{a} \right)}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}} \right) \\ & =\frac{{{\mu }_{0}}qc}{4\pi }\left( \frac{Rca-v\left( -{{c}^{2}}+{{c}^{2}}\frac{Rc}{Rc-\vec{R}\cdot \vec{v}}+{{v}^{2}}-\vec{R}\cdot \frac{Rc}{Rc-\vec{R}\cdot \vec{v}}\vec{a} \right)}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}} \right) \\ & =\frac{{{\mu }_{0}}qc}{4\pi }\left( \frac{Rca-v\left( -{{c}^{2}}+{{v}^{2}}-\frac{Rc}{Rc-\vec{R}\cdot \vec{v}}\left( -{{c}^{2}}+\vec{R}\cdot \vec{a} \right) \right)}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}} \right) \\ & =\frac{{{\mu }_{0}}qc}{4\pi }\left( \frac{Rca\left( Rc-\vec{R}\cdot \vec{v} \right)-v\left( \left( -{{c}^{2}}+{{v}^{2}} \right)\left( Rc-\vec{R}\cdot \vec{v} \right)-Rc\left( -{{c}^{2}}+\vec{R}\cdot \vec{a} \right) \right)}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}} \right) \\ & =\frac{{{\mu }_{0}}qc}{4\pi {{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left( Rca\left( Rc-\vec{R}\cdot \vec{v} \right)-v\left[ -R{{c}^{3}}+{{c}^{2}}\vec{R}\cdot \vec{v}+{{v}^{2}}Rc-{{v}^{2}}\vec{R}\cdot \vec{v}+R{{c}^{3}}-Rc\left( \vec{R}\cdot \vec{a} \right) \right] \right) \\ & =\frac{{{\mu }_{0}}qc}{4\pi {{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left( Rca\left( Rc-\vec{R}\cdot \vec{v} \right)-v\left[ \left( {{c}^{2}}-{{v}^{2}} \right)\vec{R}\cdot \vec{v}+{{v}^{2}}Rc-Rc\left( \vec{R}\cdot \vec{a} \right) \right] \right) \\ & =\frac{{{\mu }_{0}}qc}{4\pi {{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left( Rca\left( Rc-\vec{R}\cdot \vec{v} \right)-v\left( {{c}^{2}}-{{v}^{2}} \right)\vec{R}\cdot \vec{v}\,-{{v}^{3}}Rc+Rcv\left( \vec{R}\cdot \vec{a} \right) \right) \\ & =\frac{{{\mu }_{0}}qc}{4\pi {{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left( Rca\left( Rc-\vec{R}\cdot \vec{v} \right)-v\left( {{c}^{2}}+{{v}^{2}} \right)\vec{R}\cdot \vec{v}\,-{{v}^{3}}Rc+Rcv\left( \vec{R}\cdot \vec{a} \right) \right) \\ \end{align}\] \[\frac{\partial \vec{A}}{\partial t}=\frac{{{\mu }_{0}}}{4\pi }\frac{1}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left( \left( Rc-\vec{R}\cdot \vec{v} \right)\left( R\vec{a}-c\vec{v} \right)+R\left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\vec{v} \right)\]

Determinatio of E

\[\begin{align} & \frac{\partial \vec{A}}{\partial t}=\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{1}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left( \left( Rc-\vec{R}\cdot \vec{v} \right)\left( R\vec{a}-c\vec{v} \right)+R\left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\vec{v} \right) \\ & \vec{E}=-\nabla \phi -\frac{\partial \vec{A}}{\partial t} \\ & =-\frac{qc}{4\pi {{\varepsilon }_{0}}}\frac{1}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left( \left( Rc-\vec{R}\cdot \vec{v} \right)\vec{v}-\left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\vec{R} \right)-\frac{qc}{4\pi {{\varepsilon }_{0}}}\frac{1}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\frac{1}{c}\left( \left( Rc-\vec{R}\cdot \vec{v} \right)\left( R\vec{a}-c\vec{v} \right)+R\left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\vec{v} \right) \\ & =\frac{qc}{4\pi {{\varepsilon }_{0}}}\frac{1}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left( -\left( Rc-\vec{R}\cdot \vec{v} \right)\vec{v}+\left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\vec{R}-\frac{1}{c}\left( \left( Rc-\vec{R}\cdot \vec{v} \right)\left( R\vec{a}-c\vec{v} \right)+R\left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\vec{v} \right) \right) \\ & =\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{1}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left( -\left( Rc-\vec{R}\cdot \vec{v} \right)c\vec{v}+\left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)c\vec{R}-\left( \left( Rc-\vec{R}\cdot \vec{v} \right)\left( R\vec{a}-c\vec{v} \right)+R\left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\vec{v} \right) \right) \\ & =\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{1}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left( -\left( Rc-\vec{R}\cdot \vec{v} \right)\left( c\vec{v}+R\vec{a}-c\vec{v} \right)+\left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\left( c\vec{R}-R\vec{v} \right) \right) \\ & =\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{1}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left( R\vec{a}\left( \vec{R}\cdot \vec{v}-Rc \right)+\left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\left( c\vec{R}-R\vec{v} \right) \right) \\ & =\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{R}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left( \vec{a}\left( \vec{R}\cdot \vec{v}-Rc \right)+\left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\left( c\hat{R}-\vec{v} \right) \right) \\ & =\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{R}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left( -\left( Rc-\vec{R}\cdot \vec{v} \right)\vec{a}+\left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\left( c\hat{R}-\vec{v} \right) \right) \\ & =\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{R}{{{\left( \vec{R}\cdot \vec{u} \right)}^{3}}}\left( -\left( \vec{R}\cdot \vec{u} \right)\vec{a}+\left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\vec{u} \right)\left[ \forall \vec{u}=c\hat{R}-\vec{v}\Rightarrow \vec{R}\cdot \vec{u}=Rc-\vec{R}\cdot \vec{v} \right] \\ & =\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{R}{{{\left( \vec{R}\cdot \vec{u} \right)}^{3}}}\left( \left( {{c}^{2}}-{{v}^{2}} \right)\vec{u}+\left( \vec{R}\cdot \vec{a} \right)\vec{u}-\left( \vec{R}\cdot \vec{u} \right)\vec{a} \right) \\ & =\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{R}{{{\left( \vec{R}\cdot \vec{u} \right)}^{3}}}\left( \left( {{c}^{2}}-{{v}^{2}} \right)\vec{u}+\vec{R}\times \left( \vec{u}\,\times \vec{a} \right) \right) \\ & =\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{R}{{{\left( \vec{R}\cdot \vec{u} \right)}^{3}}}\left( {{c}^{2}}-{{v}^{2}} \right)\vec{u}+\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{R}{{{\left( \vec{R}\cdot \vec{u} \right)}^{3}}}\vec{R}\times \left( \vec{u}\,\times \vec{a} \right) \\ & ={{{\vec{E}}}_{coulomb\,field}}+{{{\vec{E}}}_{radiation\,field}} \\ \end{align}\]

Special case: When charge is at rest i.e \( v=0\) and \( a\) is zero

\[\begin{align} & v=0,a=0\Rightarrow \vec{u}=c\hat{R}-\vec{v}=c\hat{R},\vec{R}\cdot \vec{u}=\vec{R}\cdot c\hat{R}=cR \\ & \vec{E}=\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{R}{{{\left( cR \right)}^{3}}}\left( {{c}^{2}}-{{0}^{2}} \right)\vec{u}+\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{R}{{{\left( cR \right)}^{3}}}\vec{R}\times \left( c\hat{R}\,\times 0 \right) \\ & =\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{R}{{{\left( cR \right)}^{3}}}\left( {{c}^{2}}-{{0}^{2}} \right)c\hat{R} \\ & =\frac{q}{4\pi {{\varepsilon }_{0}}{{R}^{2}}}\hat{R} \\ \end{align}\]

When the charge is moving at constant velocity making angle \( \theta \) with distance

\[\vec{E}=\frac{q}{4\pi {{\varepsilon }_{0}}{{R}^{2}}}\left( \frac{1-\frac{{{v}^{2}}}{{{c}^{2}}}}{{{\left( 1-\frac{{{v}^{2}}{{\sin }^{2}}\theta }{{{c}^{2}}} \right)}^{3/2}}} \right)\frac{{\hat{R}}}{{{R}^{2}}}\]

Determination of \( B\)

\[\begin{align} & \vec{B}=\nabla \times \vec{A} \\ & =\nabla \times \left( \frac{{\vec{v}}}{{{c}^{2}}}\phi \right) \\ & =\nabla \times \left( \frac{{\vec{v}}}{{{c}^{2}}}\phi \right) \\ & =\frac{1}{{{c}^{2}}}\nabla \times \left( \vec{v}\phi \right) \\ & =\frac{1}{{{c}^{2}}}\left[ \vec{v}\times \nabla \phi +\phi \left( \nabla \times \vec{v} \right) \right] \\ & =\frac{1}{{{c}^{2}}}\left[ \vec{v}\times \nabla \phi +\phi \left( \vec{a}\times \nabla t' \right) \right] \\ & =\frac{1}{{{c}^{2}}}\left[ \vec{v}\times \left( \frac{qc}{4\pi {{\varepsilon }_{0}}{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left[ \left( Rc-\vec{R}\cdot \vec{v} \right)\vec{v}-\left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\vec{R} \right] \right)+\frac{qc}{4\pi {{\varepsilon }_{0}}\left( Rc-\vec{R}\cdot \vec{v} \right)}\left( \vec{a}\times \frac{-\vec{R}}{Rc-\vec{R}\cdot \vec{v}} \right) \right] \\ & =\frac{qc}{4\pi {{\varepsilon }_{0}}}\frac{1}{{{c}^{2}}}\left[ \vec{v}\times \left( \frac{1}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left[ \left( Rc-\vec{R}\cdot \vec{v} \right)\vec{v}-\left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\vec{R} \right] \right)-\left( \frac{\vec{a}\times \vec{R}}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}} \right) \right] \\ & =\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{1}{c}\left[ \left( \left[ 0-\frac{c\left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\vec{v}\times \vec{R}}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}} \right]-\left( \frac{\vec{a}\times \vec{R}}{{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{2}}} \right) \right) \right] \\ & =\frac{q}{4\pi {{\varepsilon }_{0}}{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\frac{1}{c}\left[ \left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\vec{R}\times \vec{v}-\left( \vec{a}\times \vec{R} \right)\left( Rc-\vec{R}\cdot \vec{v} \right) \right] \\ & =\frac{q}{4\pi {{\varepsilon }_{0}}{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\frac{1}{c}\left[ \left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\left( \vec{R}\times \left( c\hat{R}-\vec{u} \right) \right)-\left( \vec{a}\times \vec{R} \right)\left( \vec{R}\cdot \vec{u} \right) \right] \\ & =\frac{q}{4\pi {{\varepsilon }_{0}}{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\frac{1}{c}\left[ \left( {{c}^{2}}-{{v}^{2}}+\vec{R}\cdot \vec{a} \right)\left( 0-\vec{R}\times \vec{u} \right)-\left( \vec{a}\times \vec{R} \right)\left( \vec{R}\cdot \vec{u} \right) \right] \\ & =\vec{R}\times \frac{q}{4\pi {{\varepsilon }_{0}}{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\frac{1}{c}\left[ -\left( {{c}^{2}}-{{v}^{2}} \right)\vec{u}-\left( \vec{R}\cdot \vec{a} \right)\vec{u}+\left( \vec{R}\cdot \vec{u} \right)\vec{a} \right] \\ & =\vec{R}\times \frac{q}{4\pi {{\varepsilon }_{0}}{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\frac{1}{c}\left[ -\left( {{c}^{2}}-{{v}^{2}} \right)\vec{u}-\vec{R}\times \left( \vec{u}\times \vec{a} \right) \right] \\ & =-\hat{R}\times \frac{qR}{4\pi {{\varepsilon }_{0}}{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\frac{1}{c}\left[ \left( {{c}^{2}}-{{v}^{2}} \right)\vec{u}+\vec{R}\times \left( \vec{u}\times \vec{a} \right) \right] \\ & =-\hat{R}\times \frac{1}{c}\left( \frac{qR}{4\pi {{\varepsilon }_{0}}{{\left( Rc-\vec{R}\cdot \vec{v} \right)}^{3}}}\left[ \left( {{c}^{2}}-{{v}^{2}} \right)\vec{u}+\vec{R}\times \left( \vec{u}\times \vec{a} \right) \right] \right) \\ & =-\hat{R}\times \frac{1}{c}\vec{E} \\ \end{align}\]