Electrostatics - Electrical Conductivity

Electrical Conductivity

Current density at a point in a conductor is defined as the amount of current flowing per unit area of the conductor around that point provided the area is held in a direction normal to the current. 

Let l be the current distributed uniformly across a conductor of cross-sectional area A. the magnitude of the current density for all points on that cross- section of the conductor is 

J = I/A

Current density is a vector quantity its direction is the direction of motion of positive charge.

We know I = An e u d

∴ j = I / A = n e u d 

The unit of current density is ampere (metre)^-2 or (A m ^-2)

Note for a particular surface of conductor, the current is the flux of j over that surface s and is given by 

I = ∫ j. ds

Where ds is elementary surface area vector of as element taken over the particular surface S and integral is taken over the particular surface S and integral is taken over the surface in question.

Conductance (G) the inverse of resistance ® is called conductance of a conductor 

Conductance G = I / R

The unit of conductance is mho or sidemen (symbol S).

Electrical conductivity the inverse of resistivity (p) of a conductor is called its electrical conductivity (a)

σ = I/p

The unit of electrical conductivity is 

Mho m^-1 or Sm

The dimensions of electrical conductivity are 

σ = 1 / p = n e² τ / m = L^{– 3} x (AT)² AT/M = M^-1 L^-3 T³ A² 

Relation between j σ and E

We know I = n Ae us = n Ae (eE / m τ) = n Ae² τ E/m

Or I / A = ne² τ E / m

Or j = 1 / p E [ ∴j = 1 / A and p = m / ne² τ]

∴ J = σE

(∴σ= I / p)                                                   

Relating between resistivity and electron mobility

We know that I = n A eud ; vd = μE

And j = I / A = σE = E / p 

∴ E / p = I / A = n e u d = n e μ E 

P = 1 / ne μ 

Potential difference of 100 V is applied to the ends of a copper wire one metre long. Calculate the average drift velocity of the electrons? Compare it with thermal velocity at 27.C consider there is one conduction electron per atom. The density of copper is 9.0 x 10³ kg /m³; atomic mass of copper is 36.5 g. Avogadro’s number = 6.0x 10²³ per gram – mole. Conductivity of copper is 5.18 x 10⁷O^-1 Boltzmann constant = 1.38 x 10 ^{– 23} jk^-1

Sol:  Here; V = 100 V, I = 1m,

M = 63.5g = 63.5 x 10^-3 g;

P = 9.0 x 10³ kg / m³;

N = 6.0 x 1023 per gram – mole;

σ  = 5.81 x 107 O ^{- 1} m^-1

Since 6.1023 copper atoms have a mass of 36.5g and there is one conduction electron per atom so number density of electrons, 

N = 6.0 x 10²³ / 63.5 x 9.0 = 8.5 x 10²² cm^-3 = 8.5 x 10²⁸ m^-3

Electric filed E = V / I = 100 / 1 = 100 Vm^-1

AS J = σ E = n e vd

∴  vd = σE / ne = (5.81 x 10⁷) x (100) / (8.5 x 10²⁸) x (1. 61 x 10 ^{– 19}) = 0.43ms^-1

Thermal velocity 

V ms 3k B T / me = 3 x 1.38 x 10^-23 x 300 / 9.1 x 10^-31 = 1.17 x 10⁵ m/s

Vd / vrms = 0.43 / 1.17 x 10⁵ = 3.67 x 10^-6

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