# Basics Of Electromagnetic Theory

This post is about basic idea of Electro-magnetic theory. This post cover what is Electric field, magnetic field , how it propagate and what is strength of Electric field and magnetic field which is explained by Gauss Law and Amperes Law respectively. It also includes some idea about Gauss divergence theorem, Stocks theorem, permittivity and permeability.

## ELECTRIC FIELD

Electric field is a format of energy that is all around a charge and influences similar charges nearby.

• Stationary charge is cause of voltage (D.C Voltage) this cause have effect in form of electric field.

## MAGNETIC FIELD

Magnetic field is a format of energy that is all around a moving charge and influences similar moving charges nearby.

• Moving charge is cause of DC current which have effect in the form of magnetic field.

### OUTFLOW AND DIVERGENCE OF VECTOR FUNCTION:

Consider a cause or a source having some effects radially outworld from the cause .For all such phenomenon the strength decreases as the area of expansion increases; such that :

The total outflow through any enclosing surface is always a constant and this constant depends on the central cause”

The strength represents a density vector function or closeness of the lines ,

and mathematically

                   Strength = Constant/ Area = Cause/ Area


If a cause is of Q coulombs of charge, the effect represents the physical attractive or repulsive force on any charge nearby. This is called as Electric Field of electric flux.

The strength is called as Electric Flux Density (D) such that:

$$∯_{closed} \vec{D} \cdot \overrightarrow{d S}=\psi_{e} \alpha Q$$

Here flux is total flux which is proportional to total enclosed cause Q.

### Gauss Law in Integral form:

$$∯ \vec{D} \cdot \overrightarrow{d S}= Q$$

If the surface in not completely enclosing, the effect are partial i.e.

\begin{aligned}\iint_{O p e n} \vec{D} \cdot \overrightarrow{d S} &=\psi_{e} \\& \neq Q\end{aligned}

Here Flux passing through the surface (open), only through that open surface and this in not Gauss Law.

Gauss Law in point form:

Strength of field is given by:

So,\begin{aligned}\vec{D} &=\frac{d Q}{d S} \\\frac{d Q}{d V} &=\frac{d}{d l}\left(\frac{d Q}{d S}\right)=\nabla \cdot D \\\rho_{v} &=\nabla \cdot D\end{aligned}

Divergence at any point depends on the volume charge density. Rate of change of strength (D) depends on charge density.

## Circulation And Curl Of Vector Function:

If cause or source is I amps of current wire , then effect will be circularly around the cause and strength will be length of the line that is intensity vector function.

• As length of circulation will increases, strength will decreases and that depends on cause i.e.
• For Example Air velocity under the fan.

### Ampere Law in Integral Form:

• Cause or source : I amps of the current wire
• Effect : Circularly around the wire ( Magnetic Field/ force/flux)
• Strength : Length of the line(Intensity Vector Function)

Mathematically,

$\oint_{c lo s e d} \vec{H} \cdot \overrightarrow{d l} \alpha I$

So,

$\oint_{closed} \vec{H} \cdot \overrightarrow{d l}= I$

This is Ampere law in Integral form.

Strength around the current, ${H}=\frac{dI}{dl}$ having unit Ampere/ meter.

## Ampere Law In Point Form:

\begin{aligned}\frac{d I}{d S} &=\frac{d}{d l}\left(\frac{d I}{d l}\right) \\&=\nabla \times H \\\operatorname{also} \nabla \times H &=J\end{aligned}

This is Ampere’s Law in point form.

Intensity H depends on current density J .

### CONCLUSION 1:

• Gauss Divergence Theorem:

$∯ D \cdot d S=Q=\iiint \rho_{v} d v=\iiint \nabla \cdot D d v$

Generalized for any vector A

$∯ A \cdot d S=\iiint \nabla \cdot A d v$

Note: Any vectors effect over a closed surface can be related to the volume enclosed strictly with Divergence operation.

### CONCLUSION 2:

• Stokes Theorems:

$\oint H\cdot d l=I=\iint {J} \cdot d s=\iint (\nabla \times H)\cdot d s$

Generalized for any vector A .

$\oint A\cdot d l=\iint (\nabla \times A)\cdot d s$

Note: Any vector effect over a closed line can be related to the surface enclosed strictly with curl operation.

### CONCLUSION 3:

Every vector function divergent or curl, always has two measures of strength:

1. Intensity -per unit length measure.
2. Density -per unit area measure.
• In Electric fields, D- Coulombs/m^2 and E-Volts/m

Note: Charge (Q) also gives E field and voltage also gives E field. Hence charge is nothing but voltage only.

D and E both measure the same thing. Hence they should be proportional to each other. i.e.

$\vec{D}\alpha \vec{E}$

$\vec{D}=\in\vec{E}$

Where, $\in = \text{Material property or permitiviity to hold or allow E field to exist}$

$\in = \in_{O} \cdot \in_R ; \in_R = 1$

$\in = \in_o \text{ is the least possible value for any material}$

$\in = \in_o = \frac{1}{36\pi \times 10 ^9} F/m = \text{free space permittivity}$

• In Magnetic Fields,

$\vec{B}\alpha\vec{H}$

$\vec{B}=\mu\vec{H}$

Where,

$\mu = \text{material property or permeability to hold or allow H field to exist.}$

Also,

$\mu = \mu_o \cdot\mu_R ; \mu_R\geq 1$

$\mu_o= \text{free space permeability} = 4\pi\times 10^{-7} H/m$

Categories: Electromagnetism