Question

Two resistors A and B have resistances \[{R_A}\]​ and \[{R_B}\] respectively with \[{R_A} < {R_B}\]​. The resistivities of their materials are \[{\rho _A}\] and \[{\rho _B}\]. Then
A. \[{\rho _A} > {\rho _B}\]
B. \[{\rho _A} = {\rho _B}\]
C. \[{\rho _A} < {\rho _B}\]
D. Information insufficient

Answer 1

Hint:Before answering this question let’s understand the term resistivity. It is defined as the resistance per unit length and cross-sectional area. It depends on the length and area of the cross-section of the conductor and its unit is ohm-metre.

Formula Used:
The formula of resistance is given by,
\[R = \dfrac{{\rho \,l}}{A}\]…………. (1)
Where, \[\rho \] is the resistivity of a material, \[l\] is length of the material and \[A\] is cross-sectional area.

Complete step by step solution:
Here the value of resistance depends on the resistivity of the material, length and the area of cross-section as shown in equation (1). In the above question, they have given that the \[{R_A}\] has a resistivity of \[{\rho _A}\] and \[{R_B}\] has a resistivity. But they have not given the value of the length of the material and the cross-sectional area of that material. Then we cannot say which one has more resistivity and which one has less.

Consider that,
\[{R_A} = \dfrac{{{\rho _A}l}}{{{A_A}}}\] and \[{R_B} = \dfrac{{{\rho _B}l}}{{{A_B}}}\]
This means that they have given the \[{R_A} < {R_B}\] but may be the length of \[{R_A}\] is more and its area is less or maybe the length of \[{R_B}\] is more and its area is also large. We don’t know any information regarding the value of length and area of the two materials.

If we could have the values of length and area, we could have compared the values of both and found the answer. But now we can’t find it because as I said earlier the resistance of a material depends on the resistivity, as well as length and also area.
Therefore, the information is insufficient.

Hence, option D is the correct answer.

Note: Alloys have a significantly greater resistance than pure metals (from which they are made). Nichrome is an alloy with a high resistance. Semiconductors like germanium and silicon have resistivities that fall between those of conductors and insulators.