Question

Masses of three wires of copper are in the ratio of $1 : 3 : 5$ and their lengths are in the ratio of $5 : 3 : 1$. The ratio of their electrical resistances is

• A. 1 : 3 : 5
• B. 5 : 3 : 1
• C. 1 : 15 : 125
• D. 125 : 15 : 1

Answer 1

The Correct Option is D

To find the ratio of the electrical resistances of the three copper wires, we start by recalling the formula for resistance: R = \rho \frac{L}{A}, where \rho is the resistivity, L is the length of the wire, and A is the cross-sectional area.

Since the wires are made of copper, the resistivity \rho is constant for all three wires, so we only need to consider the ratio of the lengths and cross-sectional areas when comparing resistances. Also, remember the mass of a wire can be expressed as m = \rho \cdot V = \rho \cdot A \cdot L, where m is the mass, V is the volume, and A is the cross-sectional area.

Given that the masses of the wires are in the ratio 1:3:5 and lengths are in the ratio 5:3:1, let's denote mass and length of the wires as m_1, m_2, m_3 and L_1, L_2, L_3 respectively. Then:

\frac{m_1}{m_2} = \frac{1}{3}, \frac{m_2}{m_3} = \frac{3}{5}
\frac{L_1}{L_2} = \frac{5}{3}, \frac{L_2}{L_3} = \frac{3}{1}

We also know:

m_1 = \rho \cdot A_1 \cdot L_1, m_2 = \rho \cdot A_2 \cdot L_2, m_3 = \rho \cdot A_3 \cdot L_3

So, A_i = \frac{m_i}{\rho \cdot L_i} for i = 1, 2, 3.

The resistance for each wire can thus be expressed in terms of length and mass as:

R_i = \rho \frac{L_i}{A_i} = \rho \frac{L_i}{m_i / (\rho \cdot L_i)} = \frac{L_i^2}{m_i}

Calculating the ratios:

\frac{R_1}{R_2} = \frac{L_1^2 \cdot m_2}{L_2^2 \cdot m_1} = \left(\frac{5}{3}\right)^2 \times \left(\frac{3}{1}\right) = \frac{25}{3}

\frac{R_2}{R_3} = \frac{L_2^2 \cdot m_3}{L_3^2 \cdot m_2} = \left(\frac{3}{1}\right)^2 \times \left(\frac{5}{3}\right) = \frac{9}{5}

Which gives us \frac{R_1}{R_2} : \frac{R_2}{R_3} = 25 : 3 \times 9 : 5 = 125 : 15 : 1.

This matches with the option labeled 125 : 15 : 1, which is the correct answer. Thus, the ratio of their electrical resistances is 125 : 15 : 1.