question_answer

Two spheres of radii \[{{R}_{1}}\] and \[{{R}_{2}}\] respectively are charged and joined by a wire. The ratio of electric fields of spheres is

A)  \[\frac{R\,_{2}^{2}}{R_{1}^{2}}\]                            

B)  \[\frac{R\,_{1}^{2}}{R_{2}^{2}}\]  

C)  \[\frac{R{{\,}_{2}}}{{{R}_{1}}}\]                                

D)         \[\frac{R{{\,}_{1}}}{{{R}_{2}}}\]

Correct Answer: C

Solution :

When two spheres arc joined charge flows till it equalizes. Hence electric potential is same. \[\therefore \]  \[{{V}_{1}}={{V}_{2}}\]                 \[\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{{{q}_{1}}}{{{R}_{1}}}=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{{{q}_{2}}}{{{R}_{2}}}\]                 \[\frac{{{q}_{1}}}{{{R}_{1}}}=\frac{{{q}_{2}}}{{{R}_{2}}}\]                                               ... (i) Ratio of electric fields\[\frac{{{E}_{1}}}{{{E}_{2}}}=\frac{\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{{{q}_{1}}}{R_{1}^{2}}}{\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{{{q}_{2}}}{R_{2}^{2}}}\] \[\Rightarrow \]               \[\frac{{{E}_{1}}}{{{E}_{2}}}=\frac{{{q}_{1}}}{{{q}_{2}}}{{\left( \frac{{{R}_{2}}}{{{R}_{1}}} \right)}^{2}}\]                                    ? (ii) or            \[\frac{{{E}_{1}}}{{{E}_{2}}}=\frac{{{R}_{1}}}{{{R}_{2}}}{{\left( \frac{{{R}_{2}}}{{{R}_{1}}} \right)}^{2}}=\frac{{{R}_{2}}}{{{R}_{1}}}\]             [From Eq.(i)]