Consider a fuse wire of length $l$ and radius $r$. The time of heating $(t)$ for passing the maximum current will depend on

The heat produced by a current $I$ flowing through the fuse wire in time $t$ is given by Joule's law of heating:

$H = I^2 R t = I^2 \left(\rho \dfrac{l}{\pi r^2}\right) t$

where $\rho$ is the resistivity, $l$ is the length, and $r$ is the radius of the wire.

The heat required to raise the temperature of the fuse wire to its melting point is:

$Q = m s \Delta \theta = (\pi r^2 l d) s \Delta \theta$

where $m$ is the mass, $d$ is the density, $s$ is the specific heat, and $\Delta \theta$ is the required change in temperature.

Assuming negligible heat loss to the surroundings (adiabatic heating for a fast-blowing fuse), we equate the heat produced to the heat required:

$I^2 \left(\rho \dfrac{l}{\pi r^2}\right) t = \pi r^2 l d s \Delta \theta$

Solving for $t$, we get:

$t = \dfrac{\pi^2 r^4 d s \Delta \theta}{I^2 \rho}$

For a given current $I$, the time of heating $t$ is proportional to $r^4$ and is independent of the length $l$.

Therefore, $t \propto r^4 l^0$.

Answer: $t \propto r^4 l^0$