PPT No. 34
Principle of Conservation of Natural Quantities
Equation of continuity is
the statement of conservation of natural quantities e.g.
electric charge, energy, momentum
A conserved quantity
cannot increase or decrease,
it can only move from place to place
Principle of Conservation of Natural Quantities
The principle implies that
some natural quantities are conserved
during their transport i.e.
the net change in the total amount
(of the conserved quantity) inside any region is
equal to the resultant of the amount
that flows in and out
of the region through the boundary.
Electric Charge Conservation
(a) Global Form
Conservation of electric charge
is an important physical law in electromagnetism.
is the total charge in the Universe
Electric Charge Conservation
(a) Global Form
The net quantity of electric charge i.e.
the amount of positive charge minus
the amount of negative charge in the universe,
is always constant.
Charge can neither be created nor destroyed.
This is the global form of
Conservation of electric charge
Electric Charge Conservation
Local Form
The most important aspect of
the law of conservation of charge is that
it applies locally as well as globally.
Local form, the more powerful statement of
the law of conservation of charge may be stated as follows:
Any variation in the total charge within a closed surface
must be due to charges that flow across the surface.
Electric Charge Conservation
Local Form
The net change in the amount of electric charge
in any volume of space is exactly equal to
the resultant of amount of charge flowing
into the volume and
the amount of charge flowing out of the volume.
In essence, charge conservation states
an accounts relation between
the amount of charge in a region and
the flow of charge into and out of that same region.
Electric Charge Conservation
Local Form
Let Qin be the amount of charge
flowing into the volume between time t1 and t2
Qout is the amount of charge
flowing out of the volume during the same time period.
Q(t) is the quantity of electric charge
in a specific volume at time t, then Mathematically,
Q(t) = Q(t2) - Q(t1) = Q(in) - Q(out)
This statement of the law is called as
the continuity equation.
Continuity Equation
Continuity equation is a direct expression of
the (stronger) local form of conservation laws.
It is applicable at each point in space
& every instant in time.
It is valid in all situations and
in all frames of reference.
It is a fundamental tenet about electromagnetism.
It also characterizes realistic steady currents.
Continuity Equation
Continuity equation can be expressed in
an "integral form"
(i.e. in terms of a flux integral) and
"differential form"
(i.e. in terms of the divergence operator)
Continuity Equation
Integral Form
Electric current is defined to be
the rate of flow of charge across a surface
so according to the law of conservation of charge
where I is instantaneous current
flowing outwards through S
into the exterior space and
Q is the instantaneous charge in the enclosed volume V.
If a current flowing outwards across the surface,
there will be a loss of charge within the surface
(=>the minus sign)
Continuity Equation
Integral Form
Current I can be expressed as
a surface integral of the current density J.
Let J be the current density
(charge per unit area per unit time)
at the surface S enclosing volume V.
S = ∂V is the boundary of V and
dS is the outward pointing normal of the boundary ∂V.
Continuity Equation
Integral Form
The net current I into a volume is
The current through S is equal to
the rate of change of charge enclosed by S.
Continuity Equation
Differential Form
Using the Divergence theorem, above equation is written as
According to the principle of the conservation charge,
the net current into a volume must be equal
the net change in charge within the volume
Continuity Equation
Differential Form
Using the relation between charge and charge density ρ
This equation is valid for every volume.
It can be expressed in general as
Continuity Equation
Differential Form
The Continuity Equation is
The Continuity Equation is
a differential equation expressed
in terms of charge density ρ (in coulombs/ cubic meter)
and electric current density J (in amperes/ meter square)
It states that the divergence of the current density
is equal to the negative rate of change
of the charge density
Role of the Continuity Equation in Electromagnetism
Ampère's law in its original form is as follows.
James Clerk Maxwell found that
it was inconsistent with the conservation of charge.
He modified it as follows
∇ x B = μoJ + μoεo∂E/∂t
(in the absence of magnetic or polarizable media)
(in the presence of magnetic or polarizable media)
Role of the Continuity Equation in Electromagnetism
The Continuity equation
in the form corrected by Maxwell
became consistent with the conservation of charge.
This is one of
the four celebrated Maxwell’s equations.
Thus the principle has played an important role
in electromagnetism
to make the theory self-consistent and complete.
Continuity Equation from Maxwell’s Equations
Continuity equation can be derived
from the Maxwell’s equations as follows
Statement of the Ampère's extended law for material media
Taking the divergence of both sides
Since the divergence of a curl is zero.
LHS =0 => RHS = 0
(1)
Continuity Equation from Maxwell’s Equations
Gauss's law for material media states that
Substitution of this equation into equation (1) results in
This is the equation of continuity.
Thus Continuity equation for electric charge
can be derived using two of the Maxwell’s Equations
(Ampère's extended law and Gauss law).
Interpretation of the Continuity Equation
Current density implies flow of charge density.
The continuity equation states that
if charge is flowing out of a differential volume
(i.e. divergence of current density is positive) then
the amount of charge within that volume will decrease,
and the rate of change of charge density is negative.
Therefore the continuity equation implies
the conservation of charge.