Question

Two wires of equal length, one of copper and the other of manganin have the same resistance. Which wire is thicker?

Answer 1

Hint: Resistance of a conductor is the property by virtue of which it opposes the flow of charge through it. It depends upon several factors like the length of the conductor, area of cross-section, nature of material and temperature, etc.

Formula Used:
Resistance of a conductor is given by: $R = \rho \dfrac{l}{A}$

Complete step by step answer:
The resistance of a conductor is the property by virtue of which it opposes the flow of charge through it. It is equal to the ratio of the potential difference applied across the conductor to the current flowing through it.
At constant temperature, the resistance of a conductor depends on the following factors:
Length
The resistance R of a conductor is directly proportional to its length i.e.,
$R \propto l$

Area of cross-section
The resistance R of a uniform conductor is inversely proportional to its area of cross-section A, i.e.,
$R \propto \dfrac{1}{A}$

Nature of material
The resistance of a conductor also depends on the nature of the material it is made u of. For example, the resistance of nichrome wire is 60 times that of a copper wire of equal length and area of a cross-section.
Combining all the above expressions, we get:
$R = \rho \dfrac{l}{A}$
where $\rho $is the constant of proportionality known as resistivity or specific resistance of the material of the conductor.
Now, it is given that the length of both conductors is the same. So the resistance must be kept the same by altering the area of the cross-section only because resistivity is a material-specific property.
Let${\rho _c}{\text{ and }}{{\text{A}}_c}$ represent the resistivity and Area of cross section for copper. Similarly, let ${\rho _m}{\text{ and }}{{\text{A}}_m}$ represent the same respectively.
$\eqalign{
  & \because {R_c} = {R_m} \cr
  & \Rightarrow \dfrac{{{\rho _c}}}{{{A_c}}} = \dfrac{{{\rho _m}}}{{{A_m}}} \cr} $
But we know that ${\rho _m} > {\rho _c}$ because manganin is an alloy.
$ \Rightarrow {A_m} > {A_c}$
Therefore, between a copper wire and a manganin wire of the same length and resistance, the copper wire is thicker.

Note: Resistance of a material also depends upon the physical conditions like temperature and pressure but is independent of its shape and size. Copper due to its low resistivity is also preferred to be used to make electrical wires along with other supporting factors.