A 1 m long wire is broken into two unequal parts X and Y. The X part of the wire is stretched into another wire W. Length of W is twice the length of X and the resistance of W is twice that of Y. Find the ratio of length of X and Y.
- 1:4
- 1:2
- 4:1
- 2:1
The Correct Option is B
Approach Solution - 1
To solve this problem, we need to understand the relationship between the length and resistance of wires based on the given conditions. Here's a step-by-step explanation:
- The wire is 1 meter long, split into two parts \(X\) and \(Y\). Let \(X\) be the length of the first part and \(Y\) be the length of the second part. Hence, \(X + Y = 1 \, \text{meter}\).
- The part \(X\) is stretched into another wire \(W\) such that the length of \(W\) is twice that of \(X\). Therefore, the length of \(W\) is \(2X\).
- The resistance \(R\) of a wire is given by the formula \(R = \rho \frac{L}{A}\), where \(\rho\) is the resistivity, \(L\) is the length, and \(A\) is the cross-sectional area.
- When the length of a wire is doubled (from \(X\) to \(2X\)), its cross-sectional area \(A\) is halved to keep the volume of the wire constant, assuming uniform material. This means the resistance of \(W\) becomes four times that of \(X\), i.e., \(R_W = 4R_X\).
- According to the problem, the resistance of \(W\) is twice that of \(Y\), so \(2R_Y = 4R_X\). Thus, \(R_Y = 2R_X\).
- The resistance \(R_Y\) for wire \(Y\) is given as \(R_Y = \rho \frac{Y}{A_Y}\), and \(R_X = \rho \frac{X}{A_X}\). From \(R_Y = 2R_X\), we derive:
\(\frac{\rho Y}{A_Y} = 2 \cdot \frac{\rho X}{A_X} \)
- Simplifying, we get:
\(\frac{Y}{A_Y} = 2 \frac{X}{A_X}\)
- Assuming uniform area for both parts before stretching (i.e., \(A_Y = A_X\) before stretching), the equation simplifies to \(Y = 2X\).
- From \(X + Y = 1\) and \(Y = 2X\), solve for \(X\):
\(X + 2X = 1 \\ \Rightarrow 3X = 1 \\ \Rightarrow X = \frac{1}{3}\)
- Substitute \(X = \frac{1}{3}\) into \(Y = 2X\):
\(Y = 2 \cdot \frac{1}{3} = \frac{2}{3}\)
- The ratio of the length of \(X\) to \(Y\) is:
\(\frac{X}{Y} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2}\)
Therefore, the correct answer is 1:2.
Approach Solution -2
Rw = 2Ry
\(ρ\frac{2x}{\frac{A}{2}}=\frac{2ρ(1−x)}{A}\)
4x = 2(1 – x)
\(\frac{x}{1−x}=\frac{1}{2}\)
So, the correct option is (B): 1:2
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