Question

The electrical resistance of a mercury column in a cylindrical container is $R$. When all of the mercury is poured out into another cylindrical container of half the radius of cross section, the resistance of the new mercury column will be ______.

• A. $2R$
• B. $4R$
• C. $8R$
• D. $16R$

Answer 1

The correct option is D $16R$

Resistance is given by $R=\rho \frac{l}{A}$ where $\rho$ is the resistivity, $l$ is the length and $A$ is the area of cross section of the material.
We know that Volume of a material V = $A\times l$
$⟹l=\frac{V}{A}$
Hence $R=\rho \frac{V}{A^2}=\rho \frac{V}{{\pi }^2{r}^4}$ $\propto r^{-4}$ where $r$ is the radius of the cross section.
So according to the given conditions,
$\frac{R_1}{R_2}=\frac{(r_2{)}^4}{(r_1{)}^4}=\frac{(r/2{)}^4}{(r{)}^4}=\frac{1}{16}$
$⟹R_2=16R_1=16R$