A) 3
B) 5
C) 2
D) both [a] and [b]
Correct Answer: C
Solution :
[c] Using Gauss's law, we have \[\oint{\vec{E}.d\vec{A}}=\frac{1}{{{\in }_{0}}}\int_{{}}^{{}}{\left( \rho dv \right)=\frac{1}{{{\in }_{0}}}\int_{0}^{R}{k{{r}^{a}}\times 4\pi {{r}^{2}}dr}}\] \[\text{or }E\times 4\pi {{r}^{2}}=\left( \frac{4\pi k}{{{\in }_{0}}} \right)\frac{{{R}^{\left( a+3 \right)}}}{\left( a+3 \right)}\text{ }\therefore {{E}_{1}}=\frac{k{{R}^{\left( a+1 \right)}}}{{{\in }_{0}}\left( a+3 \right)}\]\[\text{For }r=\frac{R}{2}\cdot {{E}_{2}}=\frac{k{{\left( \frac{R}{2} \right)}^{a+1}}}{\in {{ & }_{0}}\left( a+3 \right)}=\frac{1}{8}\frac{k{{R}^{\left( a+1 \right)}}}{{{\in }_{0}}\left( a+3 \right)}\] \[\therefore {{2}^{\frac{1}{a+1}}}=\frac{1}{8}\text{ or }a=2.\]
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