# 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 1m^{2} 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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