Question

An electrical cable of copper has just one wire of radius 9mm. Its resistance is $5\Omega$. 6 different well-insulated copper wires each of radius 3 mm replace this single copper wire of the cable. 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 A $7.5\Omega$

Resistance of copper wire is
$R_c=\frac{\rho L}{A_c}$

$\rho L=R_c{A}_c=R_c\pi r_c^2$........................................$\left(1\right)$
where,
$\rho$ is resistivity of copper, $L$ is length of wire, $A_c$ is the area of cross section of wire and $r_c$ is radius of wire.

Now, single copper wire is replaced by 6 different well-insulated copper wires then the resistance of each wire is
$R=\frac{\rho L}{A}$
$\rho L=RA=R\pi r^2$ ...................................................................$\left(2\right)$

where, $r$ is radius of each wire

From $\left(1\right)$ and $\left(2\right),$ we get,

$R_c\pi r_c^2=R\pi r^2$

$R=\frac{R_c\pi r_c^2}{\pi r^2}$

$R=\frac{\left(5\right)(9{)}^2}{(3{)}^2}$ ...........................$($given$)$
$R=45\Omega$

Now, the equivalent resistance of combination is
$R_{eq}=\frac{R}{6}$

$R_{eq}=\frac{45}{6}=7.5\Omega$