A) \[\frac{\sigma }{{{\varepsilon }_{0}}}\]
B) \[\frac{\sigma }{{{\varepsilon }_{0}}}(R-r)\]
C) \[\frac{\sigma }{{{\varepsilon }_{0}}}(R+r)\]
D) None of the above.
Correct Answer: C
Solution :
Let Q and q be the charges on the spheres. The potential at the common center will be \[V=\frac{1}{4\pi {{\varepsilon }_{0}}}\times \frac{Q}{R}+\frac{1}{4\pi {{\varepsilon }_{0}}}\times \frac{q}{r}\] \[=\frac{1}{{{\varepsilon }_{0}}}\left[ \frac{Q}{4\pi {{R}^{2}}}\times R+\frac{q}{4\pi {{r}^{2}}}\times r \right]\] But \[\frac{Q}{4\pi {{R}^{2}}}=\frac{q}{4\pi {{R}^{2}}}=\sigma \] \[\therefore \]\[V=\frac{1}{{{\varepsilon }_{0}}}[\sigma R+\sigma r]=\frac{\sigma }{{{\varepsilon }_{0}}}(R+r)\] Hence, the correction option is [c].
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