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

Question

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 of P 2 : P 1 is x : 1 . The value of x is _____.

TestHub · Accepted Answer

The formula to calculate the power dissipated in the first case can be written as
$P_{1} = \frac{V_{0}^{2}}{R} . . . (1)$
When the wire is halved, its resistance will also become half the previous value.
Thus, the formula to calculate the power dissipated in the second case can be written as
$\begin{array}{rcl} P_{2} & = & \frac{V_{0}^{2}}{\frac{R}{2}} \times 2 \\ = & \frac{4 {V_{0}}^{2}}{R} . . . (2) \end{array}$
Divide equation (2) by equation (1) to calculate the required ratio.
$\begin{array}{rcl} \frac{P_{2}}{P_{1}} & = & \frac{\frac{4 {V_{0}}^{2}}{R}}{\frac{{V_{0}}^{2}}{R}} \\ = & 4 \\ = & \sqrt{16} \end{array}$
Hence, $x = 16$ .

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