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Two spherical conductors $A$ and $B$ of radii $1\, mm$ and $2\, mm$ are separated by a distance of $5\, cm$ and are uniformly charged. If the spheres are connected by a conducting wire then in equilibrium condition, the ratio of the magnitude of the electric fields at the surfaces of spheres $A$ and $B$ is-
$4 : 1$
$1 : 2$
$2 : 1$
$1 : 4$
Solution
After connection, $V_{1}=V_{2}$
$\Rightarrow K \frac{Q_{1}}{r_{1}}=K \frac{Q_{2}}{r_{2}} \Rightarrow \frac{Q_{1}}{r_{1}}=\frac{Q_{2}}{r_{2}}$
The ratio of electric fields
$\frac{E_{1}}{E_{2}}=\frac{K \frac{Q_{1}}{r_{1}^{2}}}{K \frac{Q_{2}}{r_{2}^{2}}} \Rightarrow \frac{E_{1}}{E_{2}}=\frac{Q_{1}}{r_{1}^{2}} \times \frac{r_{2}^{2}}{Q_{2}}$
$\Rightarrow \frac{E_{1}}{E_{2}}=\frac{r_{1} \times r_{2}^{2}}{r_{1}^{2} \times r_{2}} \Rightarrow \frac{E_{1}}{E_{2}}=\frac{r_{2}}{r_{1}}=\frac{2}{1}$
since the distance between the spheres is large as compared to their diameters, the induced effects may be ignored.
