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 are joined by the wire by doing so we observe that, they are at same potential and for same potential \[\frac{K{{q}_{1}}}{{{R}_{1}}}=\frac{K{{q}_{2}}}{{{R}_{2}}}\] or \[\frac{{{q}_{1}}}{{{q}_{2}}}=\frac{{{R}_{1}}}{{{R}_{2}}}\] Again the electric field at the surface of the sphere having radius R and charge q is given by \[E=\frac{Kq}{{{R}^{2}}}\] Hence \[\frac{{{E}_{1}}}{{{E}_{2}}}=\frac{\frac{K{{q}_{1}}}{R_{1}^{2}}}{\left( \frac{Kq_{2}^{2}}{R_{2}^{2}} \right)}=\frac{{{q}_{1}}}{{{q}_{2}}}\times \frac{R_{2}^{2}}{R_{1}^{2}}\] \[=\frac{{{R}_{1}}}{{{R}_{2}}}\times \frac{R_{2}^{2}}{R_{1}^{2}}=\frac{{{R}_{2}}}{{{R}_{1}}}.\]
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