ELECTROSTATICS

D. P. SINGH

DEPARTMENT OF PHYSICS
University of Petroleum & Energy Studies
Dehradun
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 CONTENTS…….
Introduction to electrostatics,
calculation of electric field,
potential and energy due to charge distribution by vector approach,
Gauss law electric flux density.
Polarization in Dielectrics,
Bound charges,
Dielectric Constant and strength,
Continuity equation and relaxation time
Boundary Conditions.

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Assume the electric field is in a vacuum or free space

Coulomb’s Law
Electrical field due to any charge configuration

Gauss’s Law
Charge distribution is symmetrical
 Coulomb’s Law

Coulomb’s law is the “law of action” between charged bodies.

Coulomb’s law gives the electric force between two point charges in
an otherwise empty universe.

A point charge is a charge that occupies a region of space which is

negligibly small compared to the distance between the point charge
and any other object.

https://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html
 Coulomb’s Law
It states that the force F between two point charges Q1 and Q2 is directly proportional
to the product of the charges and inversely proportional to the square of the distance
joining these charges and is measured in Newton.

In vector form;

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Charge is measured in coulomb and distance in meter. The constant ε0 is the vacuum
electric permittivity (also known as "electric constant") and measured in C2⋅N−1 ⋅ m−2
https://sciphy.in/electric-field-due-to-point-charge-and-system-of-charges/
 Electric Field Intensity
The electric field describes the effect of a stationary charge on other charges and is
an abstract “action-at-a-distance” concept, very similar to the concept of a gravity
field.
The basic units of electric field are Newton per Coulomb.
In practice, we usually use volts per meter.
Electric Field Intensity is the force per unit charge when placed in the Electric Field;

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In vector form

https://sciphy.in/electric-field-due-to-point-charge-and-system-of-charges/
 Electric Field due to Continuous Charge Distribution

If there is a continuous charge distribution;

say along a line, on a surface, or in a volume
Charge element dQ and total charge Q due to these charge distributions can
obtained by;
→ ℎ! "

# $→ # $ $% &!' ℎ! "

( ) → ( ) )* %+ ℎ! " ,

ρL (in C/m), ρS (in C/m2) & ρV (in C/m3) are Line, surface &
Volume charge densities.
 !-
E of a point charge is;
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The electric field intensity due to each charge distribution ρL, ρS and ρV may be
given by the summation of the field contributed by the numerous point charges
making up the charge distribution.

Replace Q by the charge element and integrate;

# ,
!- ./0 123450 !- #647380 123450
4 4

( )
!- (9:6;0 123450
4
 A Line Charge: Electric Field Intensity
Consider a line charge with uniform charge density ρL extending from A to B along the z-
axis.
The charge element dQ associated with
element dl = dz of the line is

The total charge Q is

The electric field intensity E at an arbitrary
point P (x, y, z) can be given by

The field point is generally denoted by (x, y, z) and the source point as (x’, y’, z’). So from
fig…
 Electric Field Intensity due to Line Charge

Further R may be written as;

Hence intensity equation becomes

 To evaluate this, we should define α, α1 and α2 as in given fig. ;
From triangle TPdl; Secα = R / ρ so R = ρ Secα or, ρ = R Cosα
or

Further; From triangle TPdl;

tanα = z-z’ / ρ
So, z-z’ = ρ tanα
or, z’ = z - ρ tanα
or, z’ = OT - ρ tanα
or, dz’ = - ρ sec2α d α
putting these values in last equ.
Or,

So for a finite line charge, we have;

So as a special case, for an finite line charge, if point B is at (0,0,∞) and A is

at (0,0,-∞). Then α1 = π/2 and α2 = -π/2. So z-component will vanish then
we have;
Electric Field Intensity due to Surface Charge
Consider an infinite sheet of charge in the xy plane with uniform charge density ρS.
The charge associated with an elemental area dS is

And hence total charge is

The contribution to Electric Field

at Point P (0, 0, h) by the
elemental surface is
Electric Field Intensity due to Surface Charge

Substitution of these terms in

Electric Field equation gives
Electric Field Intensity due to Surface Charge

Due to the symmetry of charge

distribution, for every element 1,
there is a corresponding element
2 whose contribution along aρ
cancels that of element 1.

So E has only z-component

In general for an infinite sheet

of charge
 Electric Flux Density http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html
Electric flux Φ is the measure of the electric field through a given surface, although an electric field in
itself cannot flow. It is a way of describing the electric field strength at any distance from the charge
causing the field.
electric flux, property of an electric field that may be thought of as the number of electric lines of
force (or electric field lines) that intersect a given area. Electric field lines are considered to originate
on positive electric charges and to terminate on negative charges.
1 q )
The electric field intensity depends on the medium in which the charges are placed. E = ar
Electric flux is ϕ = < . ?@
2
4 πε 0 r

The electric field intensity depends on the medium in which the charges
are placed.
Suppose a vector field D independent of the medium is; D = (for free
space) and the electric flux ϕ in terms of D can be defined as; ϕ = < =. ?@.
The vector field D is called the electric flux density and is measured in
coulombs per square meter.
So, All the formulas derived for E from Coulomb’s Law can be used to
calculate D, except that we have to multiply all those formulas by ε0
The flux due to the electric field E can be calculated by the general definition of flux by equations;
Q =  dQ =  ρ
L
L dl = ρ
S
S dS = ρ
V
V dV

For practical reasons, this quantity is not usually considered to be the most useful in electrostatics.
Further these quantities show that the electric field intensity is independent of the medium in which the
charges are placed (only considered free space).
ρL
For intensity due to an infinite line charge E =
2 πε 0 ρ
ρL
So, electric flux density due to an infinite line charge D =
2 πρ
For intensity due to an infinite sheet of charge
ρS )
E = an
2ε 0
So, electric flux density due to an infinite sheet of charge ρS )
D = an
2
ρ V dV )
Further, field due to the volume charge distribution E =
 4 πε 0 R 2 a r
So, electric flux density due to the volume charge distribution E = ρ V dV a)
 4π R 2 r
For the above equations for flux density, it is clear that D is a function of charge & position only. It
is independent of the medium.
 Gauss Law
This is one of the fundamental laws of electromagnetism. Gauss law provides an
easy means of finding E or D for symmetrical charge distribution such as point
charge, an infinite line charge, a spherical distribution of a charge.

It states that the total electric flux Φ through any closed surface is equal to the
total charge enclosed by that surface i.e.Φ ABCD

It states that the volume

charge density is the same as
the divergence of the electric
flux density.
 Applications; Gauss Law
Conditions
The first step to apply Gauss’s law to determine the electric field,

There should be a symmetric charge distribution.

Once it exits, then we may construct a mathematical close surface (called Gaussian
surface)

The surface is chosen in such a way, that the D is either normal or tangential to the
surface. If D is normal to the surface then; D. dS = D.dS
(as D is constant every where on the surface)

But as D is tangential to the surface then; D. dS = 0

 Applications; Gauss Law
Case I; Point Charge
 Electric Potential
The potential at any point due to a point charge Q located at
E
the origin is; )
FGHI 4

The potential at any point is the potential difference

between that point and a chosen point at which the potential
is zero.
Assuming zero potential at infinity, the potential at a distance r from the point charge
is the work done per unit charge by an external agent in transferring a test charge from
infinity to that point.
K
L
Potential difference; )JK M. Measured in Joule/Coulomb
or Volt
J
 Relationship between E and V; Maxwell’s Equation
The potential difference between points A & B is independent of the path taken i.e.
)JK )KJ .

No net work is done in moving a charge along a closed

path in an electrostatic field.

Applying Stoke's theorem;

2nd Maxwell’s equation for static electric field. (differential form)

Electric Field Intensity E is the gradient of V

Negative sign shows that direction of E is opposite to the direction
in which V increases; E is directed from higher to lower levels of V.
ELECTRIC FIELD IN MATERIALSCE

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Dielectrics
In case of non polar molecules, the centre of gravity of positive and
negative charges coincide, so these molecules do not have any permanent
dipole moment. Some common examples of non-polar molecules are H2,
N2, O2.
When a non polar molecule is placed in an electric field, the centers of
positive and negative charges get displaced and the molecules are said to
have been polarized.

Such a molecule is called induced electric dipole and its electric dipole
moment is called induced electric dipole moment.
So the polarization is a phenomenon in which an alignment of positive and
negative charges takes place with in the dielectric resulting no net increase
in the charge of the dielectric.
 Polarization in Dielectrics
Consider an atom of the dielectric consisting of an electron cloud (-Q) and a positive
nucleus (+Q).
When an electric field E is applied, the positive charge is displaced from its equilibrium
position in the direction of E by F + = Q E while the negative charge is displaced by
in the opposite direction. F− = Q E

A dipole results from the displacement of charges and the dielectric is polarized. In
polarized the electron cloud is distorted by the applied electric field.
This distorted charge distribution is equivalent to the original distribution plus the dipole
whose moment is
p = Qd
where d is the distance vector between -Q to +Q.
If there are N dipoles in a volume Δv of the dielectric, the total dipole moment due to the
electric field

For the measurement of intensity of polarization, we define polarization P (coulomb per

square meter) as dipole moment per unit volume
The major effect of the electric field on the dielectric is the creation of dipole moments
that align themselves in the direction of electric field.
This type of dielectrics are said to be non-polar. eg: H2, N2, O2
Other types of molecules that have in-built permanent dipole moments are called polar.
eg: H2O, HCl

When electric field is applied to a polar material then its permanent dipole experiences a
torque that tends to align its dipole moment in the direction of the electric field.
 Electric Dipole

An electric dipole is formed when two point charges of equal magnitude but of opposite
sign are separated by a small distance.

The potential at P (r, θ, Φ) is

If r >> d, r2 - r1 = d cosθ
and r1r2 = r2 then
But d cos θ = d . a r where d = d az

If we define p = Q d as the dipole moment, then

'
The potential dV at an external point O due to P dv

The dipole moment p is directed from –Q to +Q.

if the dipole center is not at the origin but at r ' then
 Field due to a Polarized Dielectric

Consider a dielectric material consisting of dipoles with Dipole moment P per unit volume.
'
The potential dV at an external point O due to P dv

(i)

where R2 = (x-x’)2+(y-y’)2+(z-z’)2 and R is the distance

between volume element dv’ and the point O.
But

So

Applying the vector identity

= -
Put this in (i) and integrate over the entire volume v’ of the dielectric

Applying Divergence Theorem to the first term

(ii)

where an’ is the outward unit normal to the surface dS’ of the dielectric
The two terms in (ii) denote the potential due to surface and volume charge distributions with
densities;
where ρps and ρpv are the bound surface and volume charge densities.

Bound charges are those which are not free to move in the dielectric material.

Equation (ii) says that where polarization occurs, an equivalent volume charge density, ρpv is
formed throughout the dielectric while an equivalent surface charge density, ρps is formed
over the surface of dielectric.

The total positive bound charge on surface S bounding the dielectric is

while the charge that remains inside S is

 Total charge on dielectric remains zero (as was prior to the application of an electric field).

Total charge =

We now consider the case in which dielectric contains free charge. If ρv is the free volume
charge density then the total volume charge density ρt is given by

Hence

Where
So, we may say that, the net effect of the dielectric on the electric field D is to increase E
inside it by an amount P .
The polarization would vary directly as the applied electric field.

Where χ e is known as the electric susceptibility of the material

It is a measure of how susceptible a given dielectric is to electric fields.

Dielectric Constant and Strength

We know that and
Thus

or

where ε = ε oε r

and
where є is the permittivity of the dielectric, єo is the permittivity of the free space and єr is the
dielectric constant or relative permittivity.
So, dielectric constant or relative permittivity єr is the ratio of permittivity of the dielectric
(єo) to that of free space.
No dielectric is ideal. When the electric field in a dielectric is sufficiently high then it begins to
pull electrons completely out of the molecules, and the dielectric becomes conducting.

When a dielectric becomes conducting then it is called dielectric breakdown. It depends on the
type of material, humidity, temperature and the amount of time for which the field is applied.

The minimum value of the electric field at which the dielectric breakdown occurs is called the
dielectric strength of the dielectric material.
Or

The dielectric strength is the maximum value of the electric field that a dielectric can tolerate or
withstand without breakdown.
 Continuity Equation and Relaxation Time

According to principle of charge conservation, the time rate of decrease of charge within
a given volume must be equal to the net outward current flow through the closed surface
of the volume.
The current Iout coming out of the closed surface;

(i)

where Qin is the total charge enclosed by the closed surface.

Using divergence theorem

But
Equation (i) now becomes

or (ii)

This is called the continuity of current equation. And states that there can be no accumulation of
charge at any point.
For steady current, and hence showing that the total charge leaving a
volume is the same as total charge entering it, showing the validity of Kirchoff’s law.

Effect of introducing charge at some interior point of a conductor/dielectric

According to Ohm’s law

According to Gauss’s law

Equation (ii) now becomes

or

This is homogeneous liner ordinary differential equation. By separating variables we get

Integrating both sides

 (iii)

where

Equation (iii) shows that as a result of introducing charge at some interior point of the
material there is a decay of the volume charge density ρv.
The time constant Tr is known as the relaxation time or the relaxation time.

Relaxation time is the time in which a charge placed in the interior of a material to
drop to e-1 = 36.8 % of its initial value.

For Copper Tr = 1.53 x 10-19 sec (short for good conductors)

For fused Quartz Tr = 51.2 days (large for good dielectrics)
 Boundary Conditions
If the field exists in a region consisting of two different media, the conditions that the
field must satisfy at the interface separating the media are called boundary conditions.
These conditions are helpful in determining the field on one side of the boundary when
the field on other side is known.
We will consider the boundary conditions at an interface separating;
1. Dielectric (єr1) and Dielectric (єr2)
2. Conductor and Dielectric
3. Conductor and free space
For determining boundary conditions we will use Maxwell’s equations

Where Qenc is free charge enclosed in surface. Further we have to split electric field
intensity in to two orthogonal components;
E = Et + En (tangential and normal components at interface)
 Boundary Conditions (Between two different dielectrics)

Consider the E field existing in a region consisting of two different dielectrics

characterized by є1 = є0 єr1 and є2 = є0 єr2

E1 and E2 in the media 1 and 2 can be

written as
r r r r r r
E 1 = E 1 t + E 1 n and E 2 = E 2t + E 2n

But

Assuming that the path abcda is very small with

respect to the variation in E
As Δh 0

Thus the tangential components of E are the same on the two sides of the boundary. E is
continuous across the boundary.

But

Thus

or

Here Dt undergoes some change across the surface and is said to be discontinuous across
the surface.
Applying

Putting Δh 0 gives;

Where ρs is the free charge density placed deliberately at the boundary

If there is no charge on the boundary i.e. ρs = 0 then

Thus the normal components of D is continuous across the surface.

Analogy between Electric and Magnetic Field