A coaxial cable consists of an inner wire of radius $'a'$ surrounded by an outer shell of inner and outer radii ' ${b}$ ' and '$c$' respectively. The inner wire carries an electric current is, which is distributed uniformly across cross-sectional area. The outer shell carries an equal current in opposite direction and distributed uniformly. What will be the ratio of the magnetic field at a distance ${x}$ from the axis when $(i)$ ${x}<{a}$ and $(ii)$ ${a}<{x}<{b}$ ?
JEE MAIN 2021, Diffcult
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when ${x}<{a}$
${B}_{1}(2 \pi {x})=\mu_{\circ}\left(\frac{{i}_{\circ}}{\pi {a}^{2}}\right) \pi {x}^{2}$
${B}(2 \pi {x})=\frac{\mu_{\circ} {i}_{\circ} {x}^{2}}{{a}^{2}}$
${B}_{1}=\frac{\mu_{\circ} {i}_{\circ} {x}}{2 \pi {a}^{2}}...(1)$
when ${a}<{x}<{b}$
${B}_{2}(2 \pi {x})=\mu_{0} {i}_{0}$
${B}_{2}=\frac{\mu_{{o}} {i}_{\circ}}{2 \pi {x}}...(2)$
$\frac{{B}_{1}}{{B}_{2}}=\frac{\mu_{\circ} {i}_{\circ} \frac{{x}}{2 \pi {a}^{2}}}{\frac{\mu_{\circ} {i}_{\circ}}{2 \pi {x}}}=\frac{{x}^{2}}{{a}^{2}}$
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$\left[\text { Use } \mu_0=4 \pi \times 10^{-7} \mathrm{TmA}^{-1}\right]$