Question

The resistivity of aluminium is twice that of copper and its density one third that of copper. The ratio of the resistances of aluminium to copper wires having the same mass per unit length is. (Assume both the wires are of same length)

• A. $3:2$
• B. $2:7$
• C. $2:3$
• D. $7:2$

Answer 1

The correct option is C $2:3$

Given: The resistivity of aluminium is twice that of copper and its density one third that of copper.

To find the ratio of the resistances of aluminium to copper wires having the same mass per unit length

Solution:

We know, $\text{Density}=\frac{\text{mass}}{\text{volume}}=\frac{\text{mass}}{\pi r^{2}L}⟹r^2=\frac{\text{mass}}{\pi \times L\times \text{Density}}$

And the relation ship between area of cross section and density is,

$A=\pi r^2=\pi \times \frac{\text{mass}}{\pi \times L\times \text{Density}}⟹A=\frac{\text{mass}}{L\times \text{Density}}$

hence the relationship between resistance, resistivity and density can be written as,

$R=\rho \frac{L}{A}⟹R=\rho \frac{L}{\frac{\text{mass}}{L\times \text{Density}}}⟹R=\rho \frac{L^2\times \text{Density}}{\text{mass}}$

If $R_{Al}$ and $R_{Cu}$ are the resistance per unit length of aluminium(Al) and copper(Cu} respectively. Then using the relation:

$\frac{R_{Al}}{R_{Cu}}=\frac{{\rho }_{Al}\times {\text{Density}}_{Al}}{{\rho }_{Cu}\times {\text{Density}}_{Cu}}.......\left(i\right)$ [as given the mass and length of aluminium and copper wires are same]

According to the given criteria,

${\rho }_{Al}=2{\rho }_{Cu},{\text{Density}}_{Al}=\frac{1}{3}{\text{Density}}_{Cu}⟹3\times {\text{Density}}_{Al}={\text{Density}}_{Cu}$, substituting these values in equation (i) we get the ratio as

$\frac{R_{Al}}{R_{Cu}}=\frac{2{\rho }_{Cu}\times {\text{Density}}_{Al}}{{\rho }_{Cu}\times 3\times {\text{Density}}_{Al}}⟹\frac{R_{Al}}{R_{Cu}}=\frac{2}{3}=2:3$

The ratio of the resistances of aluminium to copper wires having the same mass per unit length is $2:3$