Question

An electrical cable of copper has just one wire of radius $9$ mm. Its resistance is $5\Omega$. This single wire of 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. $7.5\Omega$
• B. $45\Omega$
• C. $90\Omega$
• D. $270\Omega$

Answer 1

The correct option is B

$45\Omega$

Step 1: Given

$R=5\Omega radius\left(r\right)=9mm$

Step 2: Calculate the resistance,

As the resistivity in terms of the resistance of wire is,

$p=\frac{RA}{L}$

$$\begin{gathered} A=initialareaL=lenghtp=resistivity \\ Firstwewillcalculate p \\ p=\frac{RA}{L} \\ =\frac{5\times \pi \times 9^2}{L} \end{gathered}$$

The resistance of the final wire combination is,

$$\begin{gathered} R_1=\frac{pL}{A\text{'}} \\ =\frac{5\times \pi \times 9^2}{L}\times \frac{L}{\pi \times 3^2} \\ =45\Omega \end{gathered}$$

Thus, the correct answer is option B.