Question

An electrical cable of copper has just one wire of radius 9 mm. Its resistance is 5 ohm. This single copper wire of the cable is replaced by 6 different well insulated copper wires each of radius 3 mm. The total resistance of the cable will now be equal to

• A. 7.5 ohm
• B. 45 ohm
• C. 90 ohm
• D. 270 ohm

Answer 1

Answer: (A)

7.5 ohm

Let the resistance of single copper wire be $R_1$ . If $\rho$ is the specific resistance of copper wire, then
$R_1=\frac{\rho \times l}{a_1}=\frac{\rho \times l}{\pi r_1^2}$ - (i)
When the wire is replaced by six wires, let the resistance of each wire be $R_2$. Then
$R_2=\frac{\rho \times l}{A_2}=\frac{\rho \times l}{\pi r_2^2}$ - (ii)
From equation (i) and (ii), we get
$\frac{R_1}{R_2}=\frac{r_2^2}{r_1^2}$ or $\frac{5}{R_2}=\frac{(3\times {10}^3{)}^2}{(9\times {10}^{-3}{)}^2};R_2=45\Omega$

These six wires are in parallel. Hence the resistance of the combination would be $R_2=7.5\Omega$