Exploring Electrical Conductivity in Metals: Deriving the Classical Theory

According to classical theory (described by drude ) generally free electrons move randomly in all possible directions and no net velocity results. If an electric field E is applied, the electrons are then accelerated with a force eE towards the anode.

So according to Newton’s Law, 

            m (dV/dt)  =  eE      ------(i)  [ as F = ma ]

Where, m = mass of electron        e = charge of electron
            E = applied E-field            V = drift velocity of electron

This electron motion will be counteracted by a frictional force ϒV  due to collisions.

Under this consideration (i) be modified as-

           m (dV/dt)  + ϒV =  eE  ------(ii)  
where, ϒ = a constant

For the steady state case (immediate situation just before collisions) we obtain :

V = Vf    Where, Vf = final drift velocity(max velocity)

dV/ dt = 0

Then (ii) reduces to           ϒ Vf  =  eE
                                     Or,    ϒ = eE / Vf

Putting this value into (ii) we get-

         m (dV/dt)  + (eE / Vf ) V =  eE  
or,     m dV     = [  eE - (eE / Vf ) V  ] dt

or    m Vf    =   [  eE - (eE / Vf ) V  ] t  
   
[ Here we use the integration range minimum to maximum . That is for V ;     0 to Vf     for t ;  0 to t   ]

or,    (m Vf2 )/ t  =  eEVf -  eEV
or,   Vf  - V =  (m Vf2 )/ eEt
or,   V    =  Vf  -   (m Vf2 )/ eEt
             =   Vf  [ 1 - (m Vf )/ eEt ] ------- (v)

In equation (v) the factor   (mVf) / eE has the unit of a time 
which is defined by   τ  = (mVf) / eE            
where, τ = relaxation time

or,  Vf =  (τ e E) / m  ---------------(vi)

Now we know that conduction current density J = Nf Vf e = σE 
Where, σ = conductivity   and Nf = number of free electron

So,      σ = ( Nf Vf e ) / E
               = ( Nf e2 τ ) / m     [ from (vi) ]
               = ( Nf e2 l ) / Vm     
Where, l = Vτ = mean free path

This is the required relation. So the conductivity is large for a large  number of free electrons and for a large relaxation time.

Relaxation time is defined as the average time between two consecutive  collisions. The distance passing by electron during relaxation time is  known as mean free path.

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Negative Resistance: Breaking Down the Basics

What is resistance?

We know that electric current is nothing but only continuous flow of charges. The electric potential (voltage) difference  established between the two terminals encourages the movement of charges. For a charge  the journey from terminal to terminal is not a direct path. Rather, it is a zigzag path that results from countless collisions with fixed atoms within the material. That is why the charge experience a hindrance to their movement.  This hindrance to the flow of charges is known as resistance. So the electric current depends on voltage and resistance. Ohm's law gives the relation among them like below.

Voltage = Current x Resistance

From this relation we see that if we increase the voltage in a resistive system, the current also increases. 

Negative Resistance:

From above discussion, we can determine that the resistance is the ratio of voltage and current. So in general for resistance in a circuit or device, if we increase the voltage, always there is  an increase in current and vice versa. But in practice, sometimes a situation arises when in a circuit or device, if we increase the voltage, the current decreases. This nature is the opposition of resistance.  That is why we describe this property as  negative resistance (opposition of resistance) but remember there is no positive resistance.

Negative resistance (NR) is a property of some electrical circuits and devices in which an increase in voltage across the device's terminals results in a decrease in electric current through it.

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