A potential is applied across a uniform wire of resistance . The power dissipation is . The wire is then cut into two equal halves and a potential of is applied across the length of each half. [2023]

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Solution

Q.
A potential $V_{0}$ is applied across a uniform wire of resistance $R$ . The power dissipation is $P_{1}$ . The wire is then cut into two equal halves and a potential of $V_{0}$ is applied across the length of each half. The total power dissipation across two wires is $P_{2}$ . The ratio $P_{2} : P_{1}$ is $\sqrt{x} : 1$ . The value of $x$ is _______. [2023]

Ans.
(16)

$P = V I = I^{2} R = \frac{V^{2}}{R}$

$Now R = \frac{ρ ℓ}{A}$

$If the wire is cut into two equal half, R = \frac{R}{2}$

$Initial P_{1} = \frac{V_{0}^{2}}{R}$

$After P_{2} = \frac{V_{0}^{2}}{R'} \times 2 \Rightarrow \frac{V_{0}^{2}}{R} \times 4$

$\frac{P_{2}}{P_{1}} = 4 = \frac{\sqrt{x}}{1} \Rightarrow x = 16$

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