Question

What happens to resistivity \[\rho \] when the area of cross section is doubled and the length is halved?

Answer 1

Hint: We are asked about the variation in the value of resistivity if the cross-sectional area of the material is doubled and the length is halved. In order to answer this question, we have to know what resistivity is and the formula of resistivity with respect to the cross-sectional area and length. Once we have done that we put the values of the new area of cross-section and length and get the new value of resistivity, thus leading us to our required answer.

Formulas used:
The formula used to find the resistivity of a material is given by,
\[\rho = \dfrac{{RA}}{l}\]
Where \[R\] is the resistance of the material, \[A\] is the area of cross-section of the material and \[l\] is the length of material.

Complete step by step answer:
We can start to answer this question by defining what resistivity is. It is the resistance offered by a material of unit cross sectional area and unit length. Let the new values of length and area be \[l'\] and \[A'\]. These new values is given as \[A' = 2A\] and \[l' = \dfrac{l}{2}\].We can find the resistivity of the material using the formula,
\[\rho = \dfrac{{RA'}}{{l'}}\]
Substituting the values, we get
\[\rho = \dfrac{{RA'}}{{l'}} \\
\Rightarrow \rho = \dfrac{{R \times 2A}}{{\left( {\dfrac{l}{2}} \right)}} \\
\therefore \rho = 4\dfrac{{RA}}{l}\]

Therefore, when the area of the cross section is doubled and the length is halved; the resistivity becomes four times the original value.

Note: Resistivity is defined as the electrical resistance of a material of unit cross-sectional area and unit length. It is a distinctive property of every material; resistivity is useful in comparing materials on the basis of their ability to conduct electric currents. High resistivity means the materials are poor conductors.