Consider the following circuit consisting of a battery, resistor and a spark gap. Let's pretend that when the pos/neg leads get close there is a spark and the spark lasts indefinitely until the battery goes dead and there is no more current for the spark.

Is there some type of math or way to tell what the temperature , emmission of the spark SHOULD be as well as the temp accross the resistor?

 

I would just use pretend math.

 

A spark is complicated. There is not a good answer. A spark changes with the air in the room, humidity, air flow, etc.
If the air is moving it will not heat up as much.

The Power in a resistor is the Voltage across the resistor X the Current in the resistor.
Heat is related to the power in subtract power out. Power in is Voltage X Current. Power out is related to air flow around the resistor, the size of the resistor and other factors.

I do not have an answer for you.

 

When we do not understand something, we use pretend physics. The 19th century invention of the luminiferous ether is a classic example. You win when other people believe you.

 

With such low voltage it is not spark, but electric arc:
https://en.wikipedia.org/wiki/Electric_arc

 

More correctly, it would be a "Plasma".
The Temperature is determined by
the amount of Current, combined with the properties of the Metal-Electrodes.
The Temperature is also affected by the various Gases surrounding the Plasma-Arc.
Adding Oxygen will seriously increase the Temperature to
the point where the Electrodes will actually get into a self-combustion condition.
Some metals, like Magnesium,
can't be extinguished once it's combustion-temperature has been reached, ( in normal Air ).
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