A) \[4\pi {{\varepsilon }_{0}}aR\]
B) \[4\pi {{\varepsilon }_{0}}a\log R\]
C) \[4\pi {{\varepsilon }_{0}}a{{R}^{2}}\]
D) \[\frac{4\pi {{\varepsilon }_{0}}a}{R}\]
Correct Answer: A
Solution :
| \[E=\frac{a\,(x\hat{i}+y\hat{j}+z\hat{k})}{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}=\frac{ar}{{{r}^{2}}}=\frac{a}{r}\cdot \hat{r}\] |
| \[d\phi =Ed\,s=\left( \frac{a}{R}\cdot \hat{r} \right)(ds\cdot \hat{r})\] |
| \[\Rightarrow \]\[\phi =\int{d\phi =4\pi aR}\] |
| \[\therefore \]\[{{Q}_{\text{enclosed}}}={{\varepsilon }_{0}}\phi =4\pi {{\varepsilon }_{0}}aR\] |
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