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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