Question

An electrical cable of copper has just one wire of radius $9 \,mm$. Its resistance is $5 \,\Omega$. This single copper wire 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. $95\,\Omega$
• D. $270\,\Omega$

Answer 1

Answer: (A)

According to given problem
Resistance of a wire $ R=\frac{\rho\,L}{\pi r^{2}}$
where L is the length of a wire r is the radius of a wire
$\rho$ is the specific resistance o f a wire
$\therefore R=\frac{\rho\,L}{\pi r^{2}}=5\, \Omega$
with $r=9 \times 10^{-3}\,m$
Now as the new radius of a single wire is
$3\times 10^{-3}\,m$ i.e., r/3 its resistance.
$R'=\frac{\rho L}{\pi\left(\frac{r}{3}\right)^{2}}=\frac{9\rho L}{\pi r^{2}}=9R $
And when 6 wires each of resistance R' are connected in parallel to form the new cable
$R_{p}=\frac{R'}{6}=\frac{1}{6}(9R)=\frac{3}{2}R$
$\therefore R_{p}=\frac{3}{2} R =\frac{3}{2} \times 5\,\Omega$
$=7.5\,\Omega$