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A solid conducting cuboid of dimensions $(a \times b \times c),$ having mass ‘m’ resistivity $ρ$ and density ‘d’ is connected through ideal battery ‘V’ volt across the opposite faces for three different situations, produces power $P_{1}, P_{2} a n d P_{3}$ respectively. Given that $P_{1} : P_{2} : P_{3} = 1 : 2 : 8$ choose the correct option. (given $a > b > c$ )

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$c = \sqrt[3]{\frac{m}{4 d}}$

$c = \sqrt[3]{\frac{m}{4 \sqrt{2} d}}$

$c = \sqrt[3]{\frac{m}{2 d}}$

$c = \sqrt[3]{\frac{3}{4 d}}$

answer is B.

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Detailed Solution

Let $a > b > c,$ so
$P_{1} = \frac{V^{2} b c}{ρ a}, P_{2} = \frac{V^{2} a c}{ρ b}, P_{3} = \frac{V^{2} a b}{ρ c}$
Volume of cuboid $= a b c = \frac{m}{d} \Rightarrow 4 \sqrt{2} c^{3} \frac{m}{d} \Rightarrow c = \sqrt[3]{\frac{m}{4 \sqrt{2} d}}$

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