A) \[Q\left[ \frac{r+R}{{{r}^{2}}+{{R}^{2}}} \right]\]
B) \[\frac{Q}{2}\left[ \frac{r+R}{{{r}^{2}}+{{R}^{2}}} \right]\]
C) \[2Q\left[ \frac{r+R}{{{r}^{2}}+{{R}^{2}}} \right]\]
D) zero
Correct Answer: A
Solution :
[a] \[{{q}_{r}}+{{q}_{R}}=Q\] ?.(1) \[\frac{{{q}_{r}}}{{{q}_{R}}}=\frac{4\pi {{r}^{2}}\sigma }{4\pi {{R}^{2}}\sigma }=\frac{{{r}^{2}}}{{{R}^{2}}}\Rightarrow \frac{{{q}_{r}}}{{{q}_{R}}}=\frac{{{r}^{2}}}{{{R}^{2}}}\] ?.(2) From (1) and (2) \[{{q}_{r}}=\frac{Q{{r}^{2}}}{{{R}^{2}}+{{r}^{2}}}\]and \[{{q}_{R}}=\frac{Q{{R}^{2}}}{{{R}^{2}}+{{r}^{2}}}\] So, \[V=\frac{{{q}_{r}}}{4\pi {{\varepsilon }_{0}}r}+\frac{{{q}_{R}}}{4\pi {{\varepsilon }_{0}}R}\] \[\Rightarrow V=\frac{Q}{4\pi {{\varepsilon }_{0}}}\left( \frac{r+R}{{{r}^{2}}+{{R}^{2}}} \right)\]You need to login to perform this action.
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