Question

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.

• A. 1:4
• B. 1:2
• C. 4:1
• D. 2:1

Answer 1

The Correct Option is B

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: 

1. 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, \(X + Y = 1 \, \text{meter}\).
2. 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 \(2X\).
3. 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.
4. 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., \(R_W = 4R_X\).
5. According to the problem, the resistance of \(W\) is twice that of \(Y\), so \(2R_Y = 4R_X\). Thus, \(R_Y = 2R_X\).
6. 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:

\(\frac{\rho Y}{A_Y} = 2 \cdot \frac{\rho X}{A_X} \)

1. Simplifying, we get:

\(\frac{Y}{A_Y} = 2 \frac{X}{A_X}\)

1. Assuming uniform area for both parts before stretching (i.e., \(A_Y = A_X\) before stretching), the equation simplifies to \(Y = 2X\).
2. From \(X + Y = 1\) and \(Y = 2X\), solve for \(X\):

\(X + 2X = 1 \\ \Rightarrow 3X = 1 \\ \Rightarrow X = \frac{1}{3}\)

1. Substitute \(X = \frac{1}{3}\) into \(Y = 2X\):

\(Y = 2 \cdot \frac{1}{3} = \frac{2}{3}\)

1. 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.

Answer 2

R_w = 2R_y
\(ρ\frac{2x}{\frac{A}{2}}=\frac{2ρ(1−x)}{A}\)
4x = 2(1 – x)

\(\frac{x}{1−x}=\frac{1}{2}\)
So, the correct option is (B): 1:2