Will Heat transfer through a metal be faster, if an electrical frequency is applied to the metal Will a metal conduct heat faster if an electrical current is applied to the that metal ? For instance if a frequency current such as the same that would go to a speaker - so like 30 hz or 50hz or 1khz  -  has there ever been studies about such a thing to prove or disprove this ?  Or if metal is vibrating - will it transfer heat better if it is physically vibrating ?  -  So the next thing it this what if the metal is vibrating at 50hz and the metal also has a current of electricity going through it at the same interval - would that metal then be able to transfer heat faster under these conditions. ? Has this been studied ? Have experiments been done ?
 A: Since charge carriers in (semi)conductors carry energy, heat flux in these materials depends on the current density, i.e. the flow rate of these carriers. Specifically, the heat flux is given by
$$ \vec q = -\kappa\vec\nabla T + \Pi \vec J $$
where $T$ is temperature, $\vec J$ is current density, $\kappa$ is thermal conductivity and $\Pi$ is the Peltier coefficient. When there is no current flow, the familiar first term tells you the rate at which heat flows from a hot region to a cold region. In addition to this effect, the second term tells you that there is also a heat flow component proportional to the current density. This term is sometimes negligible, but not always.
If you have a metal bar connected between two temperature reservoirs, one hotter ($T_H$) than the other ($T_C$), with a current $I$ flowing in the bar from the hot reservoir to the cold, the rate of heat flow in the bar is about
$$P = \frac{\kappa A}{L}(T_H-T_C) + \Pi I. $$
where $A$ is the cross section area and $L$ is the length of the bar. Note that this requires direct current, otherwise the second term will average out to zero. The Peltier coefficient is related to the Seebeck coefficient $S$ by $\Pi = TS$. Obviously the current can also cause heat to be generated in the bar, through Joule heating.
A: I am not sure whether experiments have been done on this topic. But my hypothesis for this question is as follows:

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*The electric field applied across the ends of a metal conductor causes the free electrons to flow in the opposite direction of the field. Here the metal ions do not move but vibrate a little bit due to the collision of free electrons.

*If we heat one end of the rod, then the metal atoms vibrate continuously and vibrate the neighboring atoms also, making it to flow through the metal rod.

Here if we apply the electric field across the metal conductor, then there will be a flow of free electrons. But this will not affect the heat transfer in the rod as heat transfer is based on the vibration of the metal ions.
But the inverse is applicable, a heated metal rod will have more electrical resistance to the electric field applied. Because if the temperature of a metal conductor increases, the ions of the metal vibrate more vigorously. This increases the number of collisions between the free electrons and the ions.
