PPT No. 30
* Faraday’s Law for Circuits
* Interpretation of Faraday’s emf
Integral Form of Faraday’s Law
Faraday concluded from his experiments that
a time varying magnetic field induces an electric field.
Induced electric field gives rise to emf directly proportional
to the rate of change of magnetic flux.
The line integral of the electric field E
around a circuit (closed loop) C is equal to
the negative of the rate of change
of the magnetic flux Φ
through area A enclosed by the loop
∫ E • dl = - dΦ/dt ,
where Φ is the magnetic flux.
Integral Form of Faraday’s Law
The relation between the rate of change of the magnetic
field B through the surface S enclosed by a contour C and
the electric field E along the contour is given by
where, dℓ is an infinitesimal element of the contour C.
The directions of the contour C and of dA
are related by the right-hand rule
It can further be proved that it is valid for
any closed loop in space,
and not just for conducting circuits.
Differential Form of Faraday’s Law
Electromotive force around any arbitrary closed path L
in free space
but the flux through any surface S
enclosed by the path L can be written as
Φ=
Differential Form of Faraday’s Law
In this equation the left hand side can be transformed
into a surface integral by applying Stokes’ theorem and
on the right hand side the order of
integration and differentiation can be reversed
The two integrals are over the same surface S=>
their arguments must be equal
Differential Form of Faraday’s Law
At any point in space the total E-field E
is the sum of Electrostatic field ES and
Field due to changing magnetic field EM
E = ES + EM
∇E = ∇ES + ∇EM
Because ∇ES=0
can be written as
This is the differential form of Faraday's law.
Two Forms of the Flux Rule
The Flux Rule according to Faraday’s law
has two forms as follows
In summary,
whenever the magnetic flux
through a closed loop (circuit) changes,
an emf is induced in the circuit.
This is applicable to circuits moving in magnetic field as well
Two Forms of the Flux Rule
The two forms of the Flux Rule
have exactly same meaning and
can be used interchangeably in calculations.
The two forms can be transformed into each other
by applying vector calculus theorem.
Although synonymous,
the two forms convey
different conceptual understanding
depending on the physical context.
Interpretation of Faraday’s emf
Faraday presented his experimental observations
in his paper.
Heaviside formalized and presented them
in abstract mathematical form using vectors.
His version is used as one of the four Maxwell’s equations.
Faraday’s discovery of electromagnetic induction
is considered to be a discovery of immense importance
which implies important points as follows
Induced Electric Field Different from Electrostatic field
An induced electric field E is generated by
a time-varying magnetic field B according to Faraday’s law
It is quite different in nature to
an electrostatic field due a set of stationary electric charges.
The strength of the induced electric field is
directly proportional to
the rate of change of the magnetic field.
Induced Electric Field Different from Electrostatic field
In induced electric field the electric field-lines
never begin or end, and always form closed loops
in the plane perpendicular to the magnetic field in free space.
On the contrary,
in an electrostatic field the electric field-lines
begin on positive charges, end on negative charges
Induced Electric Field Different from Electrostatic field
In the case of induced electric field,
If the magnetic field pointing
in the direction of thumb increases
then the electric field-lines circulate
in the opposite sense
to the right-hand fingers and
if the magnetic field decreases
then the electric field-lines
Circulate in the same sense
as the right-hand fingers.
Induced Electric Field Different from Electrostatic field
Induced electric field certainly can do work on
a charge which circulates in a closed loop and
thus induces emf and current in a conducting loop.
However, the electric field is generated
irrespective of
the presence of a conducting circuit.
An electrostatic field cannot do net work
on a charge circulating in a closed loop.
Induced Electric Field Different from Electrostatic field
The induced electric field is Not conservative and
the path integral along a closed path is given by Faraday’s law
∫ E • dl = - dΦ/dt ,
where Φ is the magnetic flux.
It is non-zero if the magnetic flux is time dependent.
In an inductive electric field the work done in slowly moving
a charge between two points does depend on
the path taken between the two points.
Electrostatic field does not depend on
the path taken between the two points.
It is a conservative field.
Absence of Magnetic Monopoles
Faraday’s law states that
Since Divergence of a curl = 0,
Divergence of the above equation
Therefore,
is independent of time at every point in space and
the above condition (for Divergence) is satisfied
for the assumption
Absence of Magnetic Monopoles
It implies that
Magnetic B field is always solenoidal.
It further means that
Magnetic monopoles do not exist.
Magnetic poles always occur in pairs.
Unification of the Electric and Magnetic Fields
Faraday’s law (together with Ampere’s Law)
describes the interaction of
time-varying (dynamic) electric and magnetic fields.
Faraday’s law describes
how the electric and magnetic fields are interrelated.
Faraday’s law unites the electric and magnetic fields
Unification of the Electric and Magnetic Fields
Faraday’s law implies that
The electric and magnetic fields
are not independent and
energy can flow between them
when they are time varying.
It is more appropriate to consider
these two fields as a single field
called as electromagnetic field.
Unification of the Electric and Magnetic Fields
Two Phenomena in One Equation
Faraday's law is a single equation
describing two different phenomena:
The motional emf
generated by a magnetic force on a moving wire, and
the transformer emf
generated by an electric force due to
a changing magnetic field.
Unique Rule for Two Different Phenomena
Referring to these two different aspects
of electromagnetic induction,
Richard P. Feynman,
in The Feynman Lectures on Physics
states its uniqueness as follows
"flux rule" that
the emf in a circuit is equal to
the rate of change of the magnetic flux
through the circuit applies whether
the flux changes because the field changes
or because the circuit moves (or both)....
Unique Rule for Two Different Phenomena
Richard P. Feynman notes further
Yet in our explanation of the rule
we have used two –
completely distinct laws for the two cases
for "circuit moves" and
for "field changes".
Unique Rule for Two Different Phenomena
Richard P. Feynman comments
We know of no other place in physics where
such a simple and accurate general principle
requires for its real understanding
an analysis in terms of
two different phenomena”.
From
The Feynman Lectures on Physics