A current is flowing through a thin cylindrical shell of radius $R$. If energy density in the medium, due to magnetic field, at a distance $r$ from axis of the shell is equal to $U$ then which of the following graphs is correct
Medium
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(b) When a current flows through cylindrical shell, then according to Ampere circuital law, magnetic induction inside it will be equal to zero. Hence energy density at $r < R$ is equal to zero.
Therefore, $(a)$, $(c)$ and $(d)$ are wrong.
When $r > R$, $B = \frac{{{\mu _0}i}}{{2\pi r}}$.
Since $U = \frac{{{B^2}}}{{2{\mu _0}}},$therefore, outside the shell, $U = \frac{{{\mu _0}{i^2}}}{{8{\pi ^2}{r^2}}}$. It means, just outside the shell, $U = \frac{{{\mu _0}{i^2}}}{{8{\pi ^2}{R^2}}}$ and when $r \to \infty ,\;U \to 0.$
Hence $(b)$ is correct.
Therefore, $(a)$, $(c)$ and $(d)$ are wrong.
When $r > R$, $B = \frac{{{\mu _0}i}}{{2\pi r}}$.
Since $U = \frac{{{B^2}}}{{2{\mu _0}}},$therefore, outside the shell, $U = \frac{{{\mu _0}{i^2}}}{{8{\pi ^2}{r^2}}}$. It means, just outside the shell, $U = \frac{{{\mu _0}{i^2}}}{{8{\pi ^2}{R^2}}}$ and when $r \to \infty ,\;U \to 0.$
Hence $(b)$ is correct.
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