A long circular tube of length 10m and radius 0.3 m carries a current $I$ along its curved surface as shown. A wire-loop of resistance $0.005 Ω$ and of radius 0.1m is placed inside the tube with its axis coinciding with axis of the tube. The current varies as $I = I_{0} \cos (300 t)$ where $I_{0}$ is constant. If the induced current in the loop is $N π μ_{0} I_{0}$ sin (300 t), then N is?

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Detailed Solution

Magnitude field due to current in tube is given as

$\begin{array}{l} B = \frac{μ_{0} I}{L} \\ \Rightarrow B = \frac{μ_{0} I_{0} \cos 300 t}{10} \\ m a g n e t i c f l u x t h r o u g h t h e l o o p i s g i v e n a s \\ ϕ = B A \\ \Rightarrow ϕ = \frac{μ_{0} I_{0}}{10} \times 3.14 \times 0.01 \cos 300 t \end{array}$

$\begin{array}{l} \Rightarrow ϕ 3.14 \times μ_{0} I_{0} \cos 300 t \times 10^{- 3} \\ i n d u c e d e m f i n l o o p i s g i v e n a s \\ e = - \frac{d ϕ}{d t} = 3.14 \times 300 μ_{0} I_{0} \sin 300 t \times 10^{- 3} \\ i n d u c e d c u r r e n t i n l o o p i s g i v e n a s \\ i = \frac{e}{R} = \frac{3.14 \times 300 μ_{0} I_{0} \sin 300 t \times 10^{- 3}}{0.005} \\ \Rightarrow i = 3.14 \times 10 μ_{0} I_{0} \sin 300 t \end{array}$