Question

A large insulated sphere of radius $r$ charged with $Q$ units of electricity is placed in contact with a small insulated uncharged sphere of radius $r$' and is then separated. The charge on smaller sphere will now be :

• A. $Q(r^{\prime }+r)$
• B. $Q(r+r^{\prime })$
• C. $\frac{Q}{r^{\prime }+r}$
• D. $\frac{Q{r}^{\prime }}{r^{\prime }+r}$

Answer 1

The correct option is D $\frac{Q{r}^{\prime }}{r^{\prime }+r}$

We know that the capacitance of single sphere is $C=4\pi {ϵ}_{0}r$
The capacitance of charged sphere is $C=4\pi {ϵ}_{0}r$ and capacitance of uncharged sphere is $C^{\prime }=4\pi {ϵ}_0{r}^{\prime }$
When they are placed in contact the common potential , $V=\frac{Q+Q^{\prime }}{C+C^{\prime }}=\frac{Q+0}{4\pi {ϵ}_{0}r+4\pi {ϵ}_0{r}^{\prime }}=\frac{Q}{4\pi {ϵ}_0(r+r{)}^{\prime }}$
Charge on smaller sphere of radii $r^{\prime }=C^{\prime }V=4\pi {ϵ}_0{r}^{\prime }\times \frac{Q}{4\pi {ϵ}_0(r+r{)}^{\prime }}=\frac{Q{r}^{\prime }}{r+r^{\prime }}$