A) \[a\tau +b,\frac{a{{\tau }^{2}}+b\tau }{R}\]
B) \[a\tau +b,\frac{a{{\tau }^{2}}+b\tau }{2R}\]
C) \[\frac{a\tau +b}{2},\frac{a{{\tau }^{2}}+b\tau }{R}\]
D) \[2(a\tau +b),\frac{a{{\tau }^{2}}+b\tau }{2R}\]
Correct Answer: A
Solution :
[a]: Given:\[\phi =a{{t}^{2}}+bt\] The magnitude of induced emf is \[\varepsilon =\frac{d\phi }{dt}=\frac{d}{dt}(a{{t}^{2}}+bt)=2at+b\] Current flowing,\[I=\frac{|\varepsilon |}{R}=\frac{2at+b}{R}\] Average emf\[=\frac{\int\limits_{0}^{\tau }{\varepsilon dt}}{\int\limits_{0}^{\tau }{dt}}=\frac{\int\limits_{0}^{\tau }{(2at+b)dt}}{\tau }=a\tau +b\] Total charge flowing, \[q=\int_{0}^{\tau }{Idt}=\int\limits_{0}^{\tau }{\frac{(2at+b)}{R}.dt=\frac{a{{\tau }^{2}}+b\tau }{R}}\]
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