Question 23 of 311

Two identical metallic spheres $A$ and $B$ when placed at certain distance in air repel each other with a force of $F$. Another identical uncharged sphere $C$ is first placed in contact with $A$ and then in contact with $B$ and finally placed at midpoint between spheres $A$and $B$. The force experienced by sphere $C$ will be :

A$\dfrac{3F}{2}$B$\dfrac{3F}{4}$C$F$D$2F$

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Answer: (B) $\dfrac{3F}{4}$

Let charge on $A$ and $B$ be ${q}_{A}={q}_{B}=q$.

Then force acting between them is $F=\dfrac{K{q}^{2}}{{r}^{2}}$.
When $C$ is placed in contact with $A$, charge on $A$ and $C$ will be $=\dfrac{q}{2}$
Now $C$ is placed in contact with $B$, charge on $B$ and $C$ will be $=\dfrac{q+\dfrac{q}{2}}{2}=\dfrac{3q}{4}$
Now, force experienced by sphere $C$ is

${F}^{'}={F}_{2}-{F}_{1} =\dfrac{K9{q}^{2}}{16\times \dfrac{{r}^{2}}{4}}-\dfrac{K3{q}^{2}}{8\times \dfrac{{r}^{2}}{4}} =\dfrac{\left(K\dfrac{3q}{4}-K\dfrac{q}{2}\right)}{\dfrac{{r}^{2}}{4}}\times \dfrac{3q}{4}$
$=\dfrac{3K{q}^{2}}{4{r}^{2}}=\dfrac{3F}{4}$