# Maxwell’s Equations in Point (or Differential form) and Integral form

##### Maxwell’s Equations for time-varying fields in point and Integral form are:
1. $\overrightarrow{\bigtriangledown&space;}X\overrightarrow{H}=\overrightarrow{J}+\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}$      $\Rightarrow&space;\oint_{l}\overrightarrow{H}.\overrightarrow{dl}=\oint_{s}\overline{J}.\overrightarrow{ds}+\int_{s}\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}.\overrightarrow{ds}$.
2. $\overrightarrow{\bigtriangledown&space;}X\overrightarrow{E}=-\frac{\partial&space;\overrightarrow{B}}{\partial&space;t}$       $\Rightarrow&space;\oint_{l}\overrightarrow{E}.\overrightarrow{dl}=-\int_{s}\frac{\partial&space;\overrightarrow{B}}{\partial&space;t}.\overrightarrow{ds}$ .
3. $\overline{\bigtriangledown&space;}.\overrightarrow{D}=\rho&space;_{v}$            $\Rightarrow&space;\oint_{s}\overrightarrow{D}.\overrightarrow{ds}=\int_{v}\rho&space;_{v}dv$.
4. $\overrightarrow{\bigtriangledown&space;}.\overrightarrow{B}=0$      $\Rightarrow&space;\oint_{s}\overrightarrow{B}.\overrightarrow{ds}=0$.

The 4 Equations above are known as Maxwell’s Equations. Since Maxwell contributed to their development and establishes them as a self-consistent set.  Each differential Equation has its integral part. One form may be derived from the other with the help of Stoke’s theorem (or) Divergence theorem.

##### word statements of the field Equations:-

A word statement of the field Equations is readily obtained from their mathematical statement in the integral form.

1.$\overrightarrow{\bigtriangledown&space;}X\overrightarrow{H}=\overrightarrow{J}+\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}$ $\Rightarrow&space;\oint_{l}\overrightarrow{H}.\overrightarrow{dl}=\oint_{s}\overline{J}.\overrightarrow{ds}+\int_{s}\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}.\overrightarrow{ds}$.

i.e, The magneto motive force ($\because&space;\oint_{l}\overrightarrow{H}.\overrightarrow{dl}\rightarrow$ is m.m.f)around a closed path is equal to the conduction current plus the time derivative of the electric displacement through any surface bounded by the path.

2. $\overrightarrow{\bigtriangledown&space;}X\overrightarrow{E}=-\frac{\partial&space;\overrightarrow{B}}{\partial&space;t}$$\Rightarrow&space;\oint_{l}\overrightarrow{E}.\overrightarrow{dl}=-\int_{s}\frac{\partial&space;\overrightarrow{B}}{\partial&space;t}.\overrightarrow{ds}$.

The electro motive force ($\because&space;\oint_{l}\overrightarrow{E}.\overrightarrow{dl}\rightarrow$ is e.m.f)around a closed path is equal to the time derivative of the magnetic displacement through any surface bounded by the path.

3.$\overrightarrow{\bigtriangledown&space;}.\overrightarrow{D}=\rho&space;_{v}$  $\Rightarrow&space;\oint_{s}\overrightarrow{D}.\overrightarrow{ds}=\int_{v}\rho&space;_{v}dv$.

The total electric displacement through the surface enclosing a volume is equal to the total charge within the volume.

4.    $\overrightarrow{\bigtriangledown&space;}.\overrightarrow{B}=0$  $\Rightarrow&space;\oint_{s}\overrightarrow{B}.\overrightarrow{ds}=0$.

The net magnetic flux emerging through any close surface is zero.

the time-derivative of electric displacement is called displacement current. The term electric current is then to include both conduction current and displacement current. If the time-derivative of electric displacement is called an electric current, similarly $\frac{\partial&space;\overrightarrow{B}}{\partial&space;t}$ is known as magnetic current, e.m.f as electric voltage and m.m.f as magnetic voltage.

the first two Maxwell’s Equations can be stated as

1. The magnetic voltage around a closed path is equal to the electric current through the path.
2. The electric voltage around a closed path is equal to the magnetic current through the path.
##### Maxwell’s Equations for static fields in point and Integral form are:

Maxwell’s Equations of static-fields in differential form and integral form are:

1. $\overrightarrow{\bigtriangledown&space;}&space;X\overrightarrow{H}=\overrightarrow{J}$        $\Rightarrow&space;\oint_{l}\overrightarrow{H}.\overrightarrow{dl}=\oint_{s}\overline{J}.\overrightarrow{ds}$.
2. $\overline{\bigtriangledown&space;}&space;X\overrightarrow{E}=0$           $\Rightarrow&space;\oint_{l}\overrightarrow{E}.\overrightarrow{dl}=0$.
3. $\overline{\bigtriangledown&space;}.\overrightarrow{D}&space;=&space;\rho&space;_{v}$            $\Rightarrow&space;\oint_{s}\overrightarrow{D}.\overrightarrow{ds}=\int_{v}\rho&space;_{v}dv$.
4. $\overline{\bigtriangledown&space;}.\overrightarrow{B}&space;=&space;0$             $\Rightarrow&space;\oint_{s}\overrightarrow{B}.\overrightarrow{ds}=0$.

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