E-StoreDERIVATION OF THE MOST POWERFUL LAW : COMPLETE VERSION
DERIVATION OF GENERALISED OHM’S LAW
-dipendra kr. misra
INTRODUCTION :
Most of us know the Faraday’s law of induction but it is very tedious to
apply and doesn’t hold good for open circuit so here I am giving you a
very secret formula with full derivation about which my physics teacher
told me.
I am here by presenting the proof and in my next article I shall
prove Faraday’s article from this law. This law is specially dedicated to
few guys who have continuously nudged me for giving the derivation.
{The thing written in light letter are vector quantity }
PROOF :
Now lets focus on the differential piece of conductor of length dl and in the environment surrounding the piece we have
E(electric field) : due to the deposited charge in the
resistor as well as the induced field
produced due to time varying magnetic
field.
B(magnetic field) : due to external agent
Vconductor : velocity of the differential piece
Now few things that we know from before
i = neAVd
This result will hold even here since we can stop the conductor from moving and the electron by sitting on it.
Now lets focus on any 1 electron.
It has velocity equal to
Velectron = Urandom + Vdrift + Vconductor
Now here random velocity is the velocity due to thermal
agitation.
Force on this electron is
F = eE + eVelectron×B
Here V is the V of electron.
Now the force will 3 components but we are concerned
with component along the flow of electron that is we are
concerned with the value of
F.v^d
So lets take the dot product :
F .v^d = eE .v^d + e((Urandom + Vdrift + Vconductor)×B).
v^d
We will call the LHS of second equation as F from now on
-ward.
Now a simple law of vector algebra :
(A×B).C = A.(B×C)
So the triple product in the previous equation can be cha
-nged as
F. v^d = eE. v^d – eB.((Urandom + Vdrift +Vconductor)× v^d )
Now 1 thing : -
Vdrift and v^d are parallel so Vdrift × v^d = 0
So the eqn becomes
F. v^d = eE. v^d – eB.((Urandom+Vconductor)× v^d )
Now here Vd is the average of all so by using our previou
-s knowledge we know that
<Vd> = ½ F<t>/m { t is the relaxation time }
Now in a small differential piece we have differential num
-ber of electron so
<Vd> = Vd and <t> = t
Now we need to calculate the force on all n electron in th
-e differential conductor
∑F. v^d = neE. v^d –eB.(∑(Urandom)× v^d + nVconductor× v^d)
Now ∑Urandom = 0
this is because there is no net movement of charge due
to thermal agitation.
So 
∑F. v^d = neE. v^d -
But we need to find the mean force not total force. So dividing
by n we get
∑(F/n). v^d = eE. v^d - e B.( Vconductor× v^d )
The LHS is equal to the mean force <F > . v^d
And so since we found the force we can find the drift
speed hence current.
So, using the formula for drift speed
<Vd> = ½ F<t>/m
And using our simple assumption for differential piece we get :
Vd = ½ Ft/m
Here F is the mean force :
Now taking dot product both side with v^d
We get
Vd . v^d = ½ F. v^d t /m
Now Vd and v^d are parallel so we get
Vd = ½ t/m (F. v^d)
And we already know the value of F. v^d so putting the value in
the above equation we get :
Vd = ½ t/m (eE. v^d - e B.( Vconductor× v^d ) )
Or taking e out we get
Vd = ½ te/m(E. v^d - B.( Vconductor× v^d ) )
So the current is equal to
i = neA Vd
So we get :
i =1/2 ne2 A t/m (E. v^d - B.( Vconductor× v^d ))
Now multiply and divide by dl .
And here we will use the convention. We will define
dl vector as parallel to current.
Now v^d is opposite to current so
dl = - dl v^d
Caution : while using v^d I have deliberately removed the unit
of v^d so that the previous equation is dimensionally correct.
So
We get
i = 1/2 ne2 A t/(m dl) ( E. dl v^d - B.( Vconductor× v^d dl))
So we are seeing a new term here
i =1/2 ne2 A t/(m dl)(E.dl + B.(Vconductor×dl))
(2mdl )/(ne2At) i = E.dl + B.(Vconductor×dl)
Now we know that
dR = (2mdl )/(ne2At)
so we get
i dR = E.dl + B.(Vconductor×dl)
on integrating we get
∫ i dR = ∫ E.dl + ∫ B.(Vconductor×dl)
Now we know that
E = Econservative + E nonconservative
We will represent
Econservative as Ec and E nonconservative as Enc
And so
∫ i dR = ∫ Ec .dl + ∫ Enc . dl + ∫ B.(Vconductor×dl)
This equation lacks one thing that is EMF of cell has not been
Introduced till now : so we can simply add it to get
∫ i dR = ∫ Ec .dl + ∫ Enc . dl + ∫ B.(Vconductor×dl) + ∑Ecell
This equation can be used for any circuit having any devices at
our level and can be used even when circuit is closed or not.
In this regard this equation is better than Michael Faraday’s
Law. Indeed I have always used this laws for all problem of
Electro-Magentism or even Electricity.
In potential form we know that
∫ Ec .dl = V
∫ i dR = V+ ∫ Enc . dl + ∫ B.(Vconductor×dl) + ∑Ecell
The sum :
∫ Enc . dl + ∫ B.(Vconductor×dl) is called Induced EMF.
∫ i dR = V + ∑Ecell + ∑Einduced
Now if current is same then i can be taken out of integral so
∫ i dR = i∫dR = iR
Hence in most simple form we get
iR = V + ∑Ecell + ∑Einduced
So we get all possible form of GENERALISED VERSION
OF OHM’S LAW. So never say V= IR before thinking.
And I can easily prove the Farady’s law from the above
version. Moreover the above law always holds good and
is superior to Faraday’s law.
In next article I shall prove Faraday’s law from the above law.
- blade X
27th june,2009
{THE ABOVE LAW IS SELF DERIVED AND NO PORTION IS COPIED FROM ANYWHERE }
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