- #1

Physgeek64

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## Homework Statement

A stainless-steel wire is 0.1 mm in diameter and 1 m long. If the outside of the

wire is held fixed at 20◦C, estimate the steady-state current passing through the wire

when the stainless steel at the centre of the wire begins to melt.

## Homework Equations

## The Attempt at a Solution

The solution to the heat equation will be ##T=-\frac{Hr^2}{4\kappa} +c## The ln term disappears because the temperature must be finite at the origin.

H is the source term ##H=\frac{I^2R}{Al}= \frac{I^2\rho}{A^2}## where A is the c.s.a of the wire

Since the outer surface of the wire is held at 20 degrees

##c=293+\frac{Ha^2}{4\kappa} ## where a is the radius of the wire

##T=293+\frac{Ha^2}{4\kappa}-\frac{Hr^2}{4\kappa}##

Therefore the temperature at the centre of the wire is

##T=293+\frac{Ha^2}{4\kappa}##

setting ##T=1400## which we are told is the melting point of the wire we get

##1380 = \frac{I^2\rho a^2}{\pi^2 a^4 4\kappa}##

##I=\sqrt{\frac{4(1380) \pi^2 a^2 \kappa}{\rho}}##

However this isn't right but i can't see where I've gone wrong

Many thanks