A fixed horizontal wire carries a current of $200\, A$. Another wire having a mass per unit length ${10^{ - 2}}\,kg/m$ is placed below the first wire at a distance of $2\, cm$ and parallel to it. How much current must be passed through the second wire if it floats in air without any support? What should be the direction of current in it
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For floating the second wire
$\left| \begin{array}{l}\,{\rm{Down}}\;{\rm{ward}}\;{\rm{weight}}\,\\\;\;{\rm{of}}\;{\rm{second}}\;{\rm{wire}}\end{array} \right| = \left| \begin{array}{l}\,{\rm{Magnetic}}\;{\rm{force}}\,\\\;\;\;\;\;{\rm{on}}\;{\rm{it}}\end{array} \right|$
$ \Rightarrow mg = \frac{{{\mu _0}}}{{4\pi }}.\frac{{2{i_1}{i_2}}}{a} \times l$
$ \Rightarrow \left( {\frac{m}{l}} \right)g = \frac{{{\mu _0}}}{{4\pi }}.\frac{{2{i_1}{i_2}}}{a}$
$ \Rightarrow {10^{ - 2}} \times 9.8 = {10^{ - 7}} \times \frac{{2 \times 200 \times i}}{{2 \times {{10}^{ - 2}}}} \Rightarrow i = 49\,A$
$($Direction of current is same to first wire$)$
$\left| \begin{array}{l}\,{\rm{Down}}\;{\rm{ward}}\;{\rm{weight}}\,\\\;\;{\rm{of}}\;{\rm{second}}\;{\rm{wire}}\end{array} \right| = \left| \begin{array}{l}\,{\rm{Magnetic}}\;{\rm{force}}\,\\\;\;\;\;\;{\rm{on}}\;{\rm{it}}\end{array} \right|$
$ \Rightarrow mg = \frac{{{\mu _0}}}{{4\pi }}.\frac{{2{i_1}{i_2}}}{a} \times l$
$ \Rightarrow \left( {\frac{m}{l}} \right)g = \frac{{{\mu _0}}}{{4\pi }}.\frac{{2{i_1}{i_2}}}{a}$
$ \Rightarrow {10^{ - 2}} \times 9.8 = {10^{ - 7}} \times \frac{{2 \times 200 \times i}}{{2 \times {{10}^{ - 2}}}} \Rightarrow i = 49\,A$
$($Direction of current is same to first wire$)$
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