Science

(a) State the relationship between the resistance R of a wire to its length l and cross sectional area A. Use the mathematical symbols to arrive at the final formula.
(b) Using the formula define the resistivity of a material.

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(a) The resistance ( $\text R$ ) of a wire depends on three factors :

Length ( $\text l$ ) of the wire,
Cross-sectional area ( $\text A$ ) of the wire, and
Nature (material) of the wire, expressed by a constant called resistivity ( $\text ρ$ ).

The relationship is as follows:

Resistance is directly proportional to the length of the wire.
$\text R \propto \text l$
Resistance is inversely proportional to the cross-sectional area of the wire.
$\text R \propto \dfrac{1}{\text A}$

Combining these two proportionalities :

$\text R \propto \dfrac {\text l}{\text A}$

Introducing a constant of proportionality (ρ), we get the formula:

$\text R = \text ρ\dfrac {\text l}{\text A}$

(b) From the above equation,

$\text ρ = \text R\dfrac {\text A}{\text l}$

On putting $\text l = 1\ \text m$ and $\text A = 1\ \text m^2$ in the above relation,

$\text ρ = \text R\dfrac {1}{1} \\[1em] \Rightarrow \text ρ = \text R$

So, resistivity (ρ) of a material is defined as the resistance offered by a wire of that material having unit length and unit cross-sectional area.

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