Resistance and Conductance

Resistance: The property of an object which obstruct the flow of current in a circuit is called resistance

It is measured in ohm(Ω)in SI unit.

Conduction:the reciprocal of resistance is called conductance.

$$\text{i.e., C} = \frac{\text{1}}{ \text{R}} $$

It is measured in ohm^-1(Ω^-1)or simen (S) in SI unit.

Laws of resistance in metal

The resistance of metal is directly proportional to length and inversely proportional to cross-sectional area of the metal.
i.e, R ∝ L

and R ∝ 1/A

Now, combining these relations we get:

$$\text{R} \propto \frac{\text{L}}{ \text{A}} $$

$$\text{or, R} = \frac{\rho \text{L}}{ \text{A}} $$

where, ⍴ is the constant called resistivity of the metal.

Resitivity or specific resistance (⍴)

We know that, In a metal:

$$\text{ Resistance(R)} = \frac{ \rho\text{L}}{ \text{A}} $$

$$ \rho = \frac{ \text{RA}}{ \text{L}} $$

$$ \text{If A=1 , L=1 m,  then:}$$

$$ \rho = \frac{ \text{R}}

Thus, resitivity of a metal is the resistance of the metal whose cross sectional area is 1m²  and length 1 metre. It is measured in ohm metre(Ωm) in SI unit.

Conductivity: The reciprocal of resistivity is called conductivity.

$$ \rho = \frac{1}{ \sigma} $$

It is measured in Ω^-1 or ⍴m^-1 in SI unit.

Relation between current density, electric field and conductivity: 

$\text{We know, V=IR}$

$\text{or, V} = \frac { \rho \text{L}}{ \text{ A }} \left ( \because \text {R} = \frac{\rho \text{L}}{ \text{A}}\right )$

$\text{or,} \frac { \rho \text{V}}{ \text{L}} = \frac { \text{I}}{ \text{ A } \rho } $

$\text{or, E = J.P} \left (  \because \text{E} = \frac {\text{V}}{ \text{d}} \text{and J = } \frac { \text{I}}{ \text{ A } }\right )$

$ \frac {\text{1}}{ \rho }.  \text{E = J}$

$ \therefore \text{J} \sigma = \text {E}$

This is the required relation.

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