Question

The resistance of a wire varies directly with the length and inversely as the square of the diameter. If a wire of length $20m$ and diameter $0.3cm$ has a resistance of $2ohms$, then what is the resistance of a $30m$ wire with a diameter of $0.6cm$?

Answer 1

If $R$ is the resistance, $l$ its length and $d$ its diameter, then
$R\propto \frac{1}{d^2}$ or $lk=R{d}^2$
for some constant $k$. We know $R=2$ when $l=20m$ and $d=0.3cm=\frac{0.3}{100}m$.
Hence
$k=\frac{2\times (0.3{)}^2}{20\times {10}^4}=\frac{0.09}{{10}^5}$.
If $l=30$ and $d=0.6cm=\frac{0.6}{100}m$, we get
$R=\frac{lk}{d^2}=\frac{30\times 0.09}{{10}^5}\times \frac{{10}^4}{(0.6{)}^2}=0.75ohms$.