Question

An electrical cable of copper has just one wire of radius $9mm$. Its resistance is $5\Omega$. This single copper wire is replaced by $6$ different well insulated copper wires each of radius $3mm$. 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.
$R^{\prime }=\frac{\rho L}{\pi {\left(\frac{r}{3}\right)}^2}=\frac{9\rho L}{\pi r^2}=9R$
And when 6 wires each of resistance R' are connected in parallel to form the new cable
$R_p=\frac{R^{\prime }}{6}=\frac{1}{6}(9R)=\frac{3}{2}R$
$\therefore R_p=\frac{3}{2}R=\frac{3}{2}\times 5\Omega$
$=7.5\Omega$