Question

Derive an expression for electrical conductivity of material in terms of relaxation time

Answer 1

the average velocity with which negative charge free electrons of a conductor get drifted towards the positive terminal of the battery is called Drift velocity
from the basics of a motion the final velocity of a particle moving under the effect of constant acceleration is given by
v=u+at hmmm

Answer 2

As per Newton's law we know F = ma

If E is the field intensity and e is the charge of the electron, then the force experienced by the electron is F = eE

equating above two equations

eE = ma

Here m is mass of electron

a = $\frac{eE}{m}$    ....(i)

If we assume initial velocity be 0 i.e after each collision we have assumed that electron starts from zero velocity then as per first equation of motion (v = u+at)

u = 0

v = aT

where T is mean time between collisions and v is drift velocity of electron

a =  $\frac{v}{T}$   .....(ii)

equating (i) and (ii)

we get  v =  $\frac{eET}{m}$             ....(iii)

We know current density is J = $\frac{I}{A}$     ....(iv)

Let us consider a solid metal (S) of length 'l' and area of cross-section 'A'. Let 'n' be the number of electrons per unit volume

Total charge present in the solid  Q  =  nAl(-e)    ....(v)

Then current is I = $\frac{Q}{t}$       ....(vi)

Putting (v) in (vi)

I = $\frac{n(-e)Al}{t}$      .....(vii)

and v = $\frac{l}{t}$         .....(viii)

Putting (viii) in (vii)

I = -neAv                                    .....(ix)

Putting (ix) in (iv)

J = nev                                        ....(x)

putting value of v from (iii) in (x)

we get

J = ne$\frac{eET}{m}$      ....(xi)

We know  σ = 1 /  p

p here denotes resistivity and σ is conductivity

We also know p = RA/L

where R is resistance L is length and A is area

σ = $\frac{L}{A.R}$           .....(xii)

As per Ohm's Law

R = $\frac{V}{I}$               .....(xiii)

Putting (xiii) in (xii) we get

σ = $\frac{L}{\frac{VA}{I} }$

σ = $\frac{I}{A}$ $\frac{1}{\frac{V}{L} }$

σ =  $\frac{J}{E}$

As J = I/A and E = V/L

σ = J/E

Putting value of J from (xi)

σ = $\frac{ne^{2}T }{m}$