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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.

  • Misaka's maximum electrical output is 1 billion volts (according to the toaru wiki, 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, 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, (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? (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),

    The resistance of human skin varies from person to person and fluctuates between different times of day. The 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 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):

    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?"

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    I don't think there's enough information to tell. Her output may be a billion volts, but is that open circuit voltage? Does the voltage drop when a load is applied?
    – forest
    Jan 15, 2023 at 23:55
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    I think the information is vital to knowing the answer. Many voltage-boosting techniques (such as charge pumps) only apply a high voltage with low load, so they're useful for electrostatic applications, but the voltage rapidly drops when any load is applied. You're modeling Misaka as a voltage source, but there's no reason to think she is. Without knowing such basic information about how she works, it would be impossible to make any assumptions about her capabilities.
    – forest
    Jan 16, 2023 at 0:00
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    For example, I could quite easily create a device that can generate a million volts, but you could touch the leads and only feel a faint shock. Meanwhile, a mere 120 volts could kill if the source (such as an outlet) can maintain a high current. Usually when extremely high voltages are involved (like a billion volts), the source is a current source, which tries to increase voltage until a certain current is flowing, and if the circuit is open, the voltage can get very high. However it may only be able to push 5 mA, in which case the voltage will drop very rapidly under even nominal load.
    – forest
    Jan 16, 2023 at 0:07
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    Even for an anime physics question this is pretty off topic. One has to assume many favors in regard to heat dissipation, such as environment (temp, humidity, etc.), location (open, enclosed, etc.), What the human is wearing, it's body composition, etc. Furthermore Mikoto is known for holding back, especially in these tests, as seen on her fight with Therestina Kihara Lifeline piloting the giant MAR powered suit in epi 24 of Railgun S1 (Poltergeist arc)
    – кяαzєя
    Jan 17, 2023 at 13:02
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    This question is being discussed on the meta.
    – Aki Tanaka
    Jan 18, 2023 at 5:22

1 Answer 1

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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 1010 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):

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 (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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