In an electrical cable there is a single wire of radius $9\, mm$ of copper. Its resistance is $5\,\Omega $. The cable is replaced by $6$ different insulated copper wires, the radius of each wire is $3\,mm$. Now the total resistance of the cable will be ............... $\Omega$
Diffcult
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(a) Initially : Resistance of given cable
$R = \rho \frac{l}{{\pi \times {{(9 \times {{10}^{ - 3}})}^2}}}$ ... $(i)$
Finally : Resistance of each insulated copper wire is
$R\,' = \rho \frac{l}{{\pi \times {{(3 \times {{10}^{ - 3}})}^2}}}$. Hence equivalent resistance of cable ${R_{eq}} = \frac{{R\,'}}{6} = \frac{1}{6} \times \left( {\rho \frac{l}{{\pi \times {{(3 \times {{10}^{ - 3}})}^2}}}} \right)$….$(ii)$
On solving equation $(i)$ and $(ii)$ we get $R_{eq} = 7.5 \,\Omega$
$R = \rho \frac{l}{{\pi \times {{(9 \times {{10}^{ - 3}})}^2}}}$ ... $(i)$
Finally : Resistance of each insulated copper wire is
$R\,' = \rho \frac{l}{{\pi \times {{(3 \times {{10}^{ - 3}})}^2}}}$. Hence equivalent resistance of cable ${R_{eq}} = \frac{{R\,'}}{6} = \frac{1}{6} \times \left( {\rho \frac{l}{{\pi \times {{(3 \times {{10}^{ - 3}})}^2}}}} \right)$….$(ii)$
On solving equation $(i)$ and $(ii)$ we get $R_{eq} = 7.5 \,\Omega$
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