Question

A conducting sphere of radius $10\text{cm}$ is charged with $10\text{C}$. Another uncharged sphere of radius $20\text{cm}$ is allowed to touch it for some time. After that if the spheres are separated, then surface density of charges on the spheres will be in the ratio of

• A. $1:4$
• B. $1:3$
• C. $2:1$
• D. $1:1$

Answer 1

The correct option is C $2:1$

When two conducting spheres touch each other, there will be a flow of charge until they both have the same potential.
Let, $R_1$ and $R_2$ be the radii of spheres 1 and 2 respectively. Let, $Q_1$ and $Q_2$ be the charges on sphere 1 and 2 respectively, after they are seperated since, both have same potential.
$V=\frac{Q_1}{C_1}=\frac{Q_2}{C_2}$
$\Rightarrow \frac{Q_1}{4\pi {ϵ}_0{R}_1}=\frac{Q_2}{4\pi {ϵ}_0{R}_2}$
$\Rightarrow \frac{Q_1}{Q_2}=\frac{R_1}{R_2}$
Surface charge density on sphere 1 is
${\sigma }_1=\frac{Q_1}{4\pi R_1^2}$
Surface charge density on sphere 2 is
${\sigma }_2=\frac{Q_2}{4\pi R_2^2}$
$\Rightarrow \frac{{\sigma }_1}{{\sigma }_2}=\frac{Q_1}{4\pi R_1^2}\times \frac{4\pi R_2^2}{Q_2}=\frac{Q_1}{Q_2}\times \frac{R_2^2}{R_1^2}$
$\Rightarrow \frac{{\sigma }_1}{{\sigma }_2}=\frac{R_1}{R_2}\times \frac{R_2^2}{R_1^2}$
$\Rightarrow \frac{{\sigma }_1}{{\sigma }_2}=\frac{R_2}{R_1}=\frac{20}{10}$
$\Rightarrow \frac{{\sigma }_1}{{\sigma }_2}=\frac{2}{1}$

${\sigma }_1:{\sigma }_2=2:1$
$\therefore$ (C).