67662


A resistor produces/dissipates heat when a voltage drop is applied. A human
body has the property of electrical resistance.

How much heat dissipation would occur (or could theoretically occur) if Misaka
used her ability to the max on a regular human of similar build to herself?

### My attempt at setting up the math

For a compilation of information about Misaka's abilities, see [the dedicated
page on the Toaru wiki
site](https://toarumajutsunoindex.fandom.com/wiki/Misaka_Mikoto/Abilities).

  * Misaka's maximum electrical output is 1 billion volts ([according to the toaru wiki](https://toarumajutsunoindex.fandom.com/wiki/Misaka_Mikoto/Abilities#:%7E:text=Being%20the%20third%2Dranked%20Level,output%20of%201%20billion%20volts.), citing the Railgun manga's first chapter and the light novel's second volume's first chapter's second part)

  * I'm not aware of any stated limits on her ability to induce electrical currents, but according to [the Toaru wiki page on the Railgun ability](https://toarumajutsunoindex.fandom.com/wiki/Electromaster#Railgun_\(special_move\)), Misaka's application of the ability uses approximately 2,250,000 amps, citing the Index light novel Volume 20 Chapter 3 Part 7. Quoting from the Toaru wiki page:

> The third-ranked Level 5 is able to fire a metal projectile at three times
> the speed of sound by utilizing a great amount of electricity, supposedly
> somewhere in the region of 2,250,000 amperes, or about 1,000,000 volts,
> 1/10th of her power.

Note: I personally don't quite understand the logic/calculation with the amps,
volts, and power in that quote. Maybe I'm missing some context.

  * According to [Joule Heating](https://en.wikipedia.org/wiki/Joule_heating), (quoting wikipedia,) " _the power of heating generated by an electrical conductor equals the product of its resistance and the square of the current:`P = I^2 R`._"

For direct current (assuming that's the natural first choice for Misaka's
abilities (though I don't doubt she could use alternating current)), one can
use `P = I (V_A - V_B)` and `P = V^2 / R`.

  * The specific heat formula says `Q = m c delta_T`, which can be rearranged to `delta_T/delta_t = P/(m c)`.

  * According to [Which is the specific heat of human body?](https://chemistry.stackexchange.com/q/44493/129302) (A Q&A on our [chem.se] site), a good rough estimate for the specific heat of an average human body is `3.3 kJ/(kg K)`.

  * [According to wikipedia (Electrical_injury#Body_resistance)](https://en.wikipedia.org/wiki/Electrical_injury#Body_resistance),

> The resistance of human skin varies from person to person and fluctuates
> between different times of day. The
> [NIOSH](https://en.wikipedia.org/wiki/NIOSH) states "Under dry conditions,
> the resistance offered by the human body may be as high as 100,000 ohms. Wet
> or broken skin may drop the body's resistance to 1,000 ohms," adding that
> "high-voltage electrical energy quickly breaks down human skin, reducing the
> human body's resistance to 500 ohms".

  * [The Toaru wiki page on Misaka](https://toarumajutsunoindex.fandom.com/wiki/Misaka_Mikoto) doesn't seem to have information on her body mass, but it does say that she is 161cm in height, and [According to wikipedia (Human_body_weight#Ideal_body_weight)](https://en.wikipedia.org/wiki/Human_body_weight#Ideal_body_weight):

> The most common estimation of IBW is by the Devine formula [...]
>
> Female ideal body weight = 45.5 kilograms (100 lb) + 0.9 kilograms (2.0 lb)
> × (height (cm) − 152)

I've provided a self-answer based on my own (limited) knowledge and
understanding of physics and the Toaru universe. Please consider providing a
better answer if you are equipped to do so!

Note: If you're wondering what practical purpose this question serves, it's as
a precursor to [" _How does Misaka survive heat dissipation in her own body
when using her abilities?_ "](https://anime.stackexchange.com/q/67668/70615)


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a-certain-scientific-railgun
How much heat damage can Misaka's shocks do to a regular human at max output?


Disclaimer: I have a very limited knowledge and understanding of physics and
the Toaru universe. If you are equipped to give a better answer, please
consider doing so.

The following calculation is based on modeling Misaka as a voltage source
instead of a current source. I have no idea if that's consistent with canon.

The above formulae can be arranged to give `delta_T/delta_t = V^2 / (R m c)`,
where

  * `delta_T` is change in temperature
  * `delta_t` is change in time
  * `V` is 1 billion volts
  * `R` is a reasonable resistance value for the human body (I'll use the 500Ω value that occurs when human skin breaks down due to high voltage)
  * `m` is an estimate of Misaka's body mass, which can be calculated by her height and the Devine formula to be `53.6kg`
  * `c` is a reasonable estimate for the human body's specific heat, which is `3.3 kJ/(kg K)`

That would be (unless I've made any stupid errors) `(10^9 V)^2 / (500 Ω * 53.6
kg * 3.3*10^3 J/(kg K))`, which equals 11307100859.33966531 Kelvin/second (or
alternatively, degrees Celcius per second).

In scientific notation, that's **1.13071 x 10 10 Kelvin/second**.

Just multiply that by one second to get the change in temperature after one
second.

For reference ([from wikipedia, the temperature of the sun
is](https://en.wikipedia.org/wiki/Sun)):

> Center (modeled): 1.57×10^7 K  
>  Photosphere (effective): 5,772 K

So if the physical matter of the poor hypothetical soul whose similar-in-
build-to-Misaka, shocked-by-Misaka-at-maximum-output body could somehow remain
intact for 1.388508 milliseconds, that's how long it would take for it to
reach the (modeled) temperature of the center of the sun.

The above calculations do not take into account that [a material's specific
heat can vary with
temperature](https://physics.stackexchange.com/a/706723/351695) (although at
high temperatures, the variation is minimal, so that may not have any dramatic
effect here), and of course, the already-stated (from the wikipedia quote)
fact that the electrical resistance of the human body varies with temperature-
especially at extremes. If we were considering the material to get incinerated
(as it would in reality, instead of magically staying intact), then we'd want
to account for which materials burn away first (Ex. I'm guessing fat and
muscle would go before the bones).

But regardless, if you're planning to make Misaka angry, you'd better bring a
Touma with you.


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