how to derive the equation - resistivity = VA/LI
Answers
There is actually a student-friendly microscopic model how to derive the real Ohm's law
j⃗ =σE⃗ .
After its derivation you can transform it into the more common form using the answer by Nesp.
The idea goes as following:
We must start with the definition of current:
I=ΔQΔt.
So where does current come from? Current is the result of movement of charged particles in the material. Obviously, current will be proportional to the charge of one particle, the speed of the particle and the total number of particles. Current density j⃗ can therefore be written as
j⃗ =Nqv⃗ d,
where N is the density of particles, q is the charge of one particle and v⃗ d is the drift speed, that is average speed of particles. I think that this definition is self-explanatory, but it can also be derived more strictly from the second formula.
In material you have certain amount of almost "free" electrons. Those electrons behave like particles in the gas, they are crashing between themselves and into atom cores, bouncing back and forth and there is actually no net movement, average speed and current are zero.
However, if you put some potential on the ends of material, you actually put homogenous electric field in the material, which strength is
E=Vl
All electrons start accelerating in the direction of the positive potential and you can easily obtain this acceleration using the expression
a⃗ =F⃗ m=qE⃗ m.
So you actually get net movement of electrons. And now comes the beauty of Ohms law. You should ask yourself: If electrons accelerate, how come current (which is proportional to average speed of electrons) isn't becoming larger and larger with the time?
The reason is that electrons keep crashing into atom cores and after those crashes their speed is by average reset back to zero! So let's define some typical time between two crashes τ. The average maximum speed of electrons between two crashes shall be
v⃗ max=a⃗ τ=qE⃗ mτ.
Obviously, average speed between two crashes, which equals drift speed is half of that value.
From the definition of current density you finally obtain
j⃗ =Nq2τ2mE⃗
which is Ohm's law.
Therefore - and this is direct answer to your question - conductivity is
σ=1ρ=Nq2τ2m
Answer:
wrong...................