Question

A fuse wire of radius $0.1mm$ melts when a current of $10A$ is passed through it. Find the current at which a fuse wire of $0.12mm$ will melt.

Answer 1

The resistance $R=\rho \frac{l}{A}$

Cross sectional area of the wire $A=\pi r^2$

Given, for radius $r_1=0.1mm,$ melting current $I_1=10A$

Now, for radius $r_2=0.12mm,$ melting current $I_2=?$

By Joule's law of heating,

Heat produced in a wire is $H=I^{2}Rt\Rightarrow H=I^2\left(\rho \frac{l}{\pi r^2}\right)t$

We assume in both case fuse wire has same material and length. Thus $\rho ,l$ are constant and for melting in both cases they need same heat $H$. Time $t$ is also assumed same.

$\Rightarrow H\pi r^2=\rho l{I}^{2}t$

$\Rightarrow I^2\propto r^2$

Thus, $I\propto r$

so, $\frac{I_2}{I_1}=\frac{r_2}{r_1}=\frac{0.12}{0.1}=1.2$

or $I_2=1.2I_1=1.2\times 10=12A$

Answer: the current at which a fuse wire of $0.12mm$ will melt is $12A.$

will melt