A) \[\sqrt{\frac{2}{5}}{{V}_{0}}\]
B) \[\sqrt{\frac{3}{5}}{{V}_{0}}\]
C) \[\sqrt{2}{{V}_{0}}\]
D) \[\sqrt{3}{{V}_{0}}\]
Correct Answer: B
Solution :
[b] \[V_{rms}^{2}=\,\,<{{V}^{2}}>\,=\frac{V_{1}^{2}+V_{2}^{2}+V_{3}^{2}+.......}{N}\] \[=\frac{\int_{{}}^{{}}{{{V}^{2}}dN}}{\int_{{}}^{{}}{dN}}\text{ here }\frac{dN}{dV}=N\left( V \right)\] \[V_{rms}^{2}=\frac{1}{N}\int\limits_{0}^{\infty }{N\left( V \right){{V}^{2}}dV}\] \[=\frac{1}{N}\int\limits_{0}^{{{V}_{0}}}{\left[ \frac{3N}{V_{0}^{3}}{{V}^{2}} \right]{{V}^{2}}dV=\frac{3}{5}V_{0}^{2}}\] \[\Rightarrow {{V}_{rms}}=\sqrt{\frac{3}{5}}{{V}_{0}}\]You need to login to perform this action.
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