Ductile Resistor

A hypothetical metal is infinitely ductile, meaning that it can be elongated forever without breaking. At time $t = 0$ , a piece of this metal begins in the shape of a circular cylinder with radius $r_0$ and length $L_0$ .

The cylinder's length then increases at a constant rate of $\alpha$ while the volume remains constant. It maintains a cylindrical shape at all times.

An ideal DC voltage source remains connected across the cylinder.

The total energy $($ starting at $t=0)$ dissipated in the resistor due to Ohmic losses asymptotically approaches a particular value as time advances toward infinity.

What is this value in $\text{MJ}$ (megajoules)?

Details and Assumptions:

• $r_0 = 1 \ \text{cm}.$
• $L_0 = 10 \ \text{cm}.$
• Resistivity $\rho = 2 \times 10^{-8} \, \Omega\, \text{m}.$
• $\alpha = 1 \ \text{cm/s}.$
• The DC voltage is $10$ volts.
• Neglect exotic phenomena, such as transient inductances (due to dynamic material deformation) and speed-of-light issues.