Question 3 Solution: Thomson Effect

The asymmetry in the temperature distribution is caused by the Thomson Effect. Unlike Joule heating ($I^2R$), which depends only on the magnitude of current, the Thomson effect depends on the direction of current relative to the temperature gradient.

The rate of heat generation per unit volume ($\dot{q}$) in a conductor carrying current density $J$ is given by:

Where:

• $\rho J^2$: Always positive (heating). Symmetric about the center.
• $\mu$: The Thomson Coefficient (material property).
• $dT/dx$: The temperature gradient.

Because the term $-\mu J \frac{dT}{dx}$ changes sign when the gradient $dT/dx$ flips (from uphill to downhill), it breaks the symmetry of the heat distribution.

The direction of the shift depends on the sign of $\mu$ for the specific metal:

• Positive Thomson Effect ($\mu > 0$): (e.g., Copper, Antimony, Silver)
  Heat is absorbed when current flows $Cold \to Hot$ and evolved when flowing $Hot \to Cold$.
  Result: The “heat” appears to flow with the conventional current. The peak shifts downstream (towards D).
• Negative Thomson Effect ($\mu < 0$): (e.g., Iron, Platinum, Bismuth)
  Heat is evolved when current flows $Cold \to Hot$.
  Result: The “heat” moves against the conventional current (with the electrons). The peak shifts upstream (towards C).