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How do I precisely calculate the Current during a short-circuit event?

Physics Asked by hyacinth chan on December 20, 2020

I am curious about this problem due to an observation last week. A standing fan I had has its wires exposed, then it short-circuited since the neutral wire was touching the live wire directly.

Then I hear a loud "pop!" with a flash of orange light.

After my dad fixed it I started to wonder: what happened during the event? Can I calculate the electric current during the process?

So I started working with what I know.

First, there was an orange flash. Now due to electrons emit different wavelengths of lights according to their energy, I can indirectly use this information to calculate its momentum as a matter-wave. So:

begin{equation}
p=h/lambda
end{equation}

So I can use it to, once again, using the matter-wave property of an electron, calculate its velocity using:

begin{equation}
p=gamma m_ev
end{equation}

Now here comes my first bottleneck, do I need to use its relativistic form or the Newtonian form? Does the end product mean the drift velocity? Still, I used the relativistic form and pressed on, and assumed that it gives me the drift velocity. The answer is that the electrons have a drift velocity of about 1km/s. A bit too high but since this is a short-circuit so I would assume that it was normal.

Then I just input it and the constants for electron density and wire area into the General Current formula:

begin{equation}
I=NAvq
end{equation}

The end product says that the current is about 8 million amperes!

Although I am still happy to obtain a result at least, I want to know if there is a better method though. Intuitively my method feels wrong but I cannot think of anything else.

3 Answers

The color of the light in the discharge isn’t related to the velocity of the electrons in the wire. It’s related to how hot the ( ionized) air at the contact became when the wires touched.

To use this information to determine the current through the short you’d need to know not just the color of the flash, but also how much material was being heated (and the materials heat capacity) and possibly what cooling mechanisms were mitigating the heating.

If part of the flash comes from vaporizing the wire material, then the color might depend mostly on that material rather than on the current ( I.e., Cooper will simply emit a characteristic color when vaporized, regardless of the current).

Correct answer by The Photon on December 20, 2020

When dealing with short-circuits, you can have what is called an arcing fault, or a bolted fault.

An arcing fault can result in very high currents for short durations where the fault is often cleared by the localized melting of the contacting conductors before an overcurrent protection device can detect and interrupt it. Calculating the magnitude of an arcing fault current and flash energy, which is of concern from a fire hazard safety standpoint, can be very complicated. The following link provides examples for flash energy and fault current:

https://www.ecmweb.com/content/article/20891596/calculating-arc-flash-energy-levels

A bolted fault is one that does not clear at the fault but is interrupted by an overcurrent protection device, typically a circuit breaker or a fuse. You can find published curves giving you the magnitude and duration of the fault current based on the operating characteristics of the overcurrent protection device.

The following link provides examples of such curves for various types of fuses: https://www.eaton.com/Eaton/ProductsServices/Electrical/Support/Documentation/TimeCurrentCurves/Fuses/index.htm

Finally, the following link provides an explanation and example of time-current characteristics of circuit breakers:

https://electrical-engineering-portal.com/time-current-curves

Hope these can be of help.

Answered by Bob D on December 20, 2020

The current during short-circuit varies in time. Its maximum value in your case would ideally be limited by circuit-breakers on your power line, in houses/appartments usually in range of 10-20A. For extremely short period of time, current can reach greater value than that, as the space between phase and neutral becomes very conductive. Exact value is hard to calculate because it depends on geometry and electric properties of the metals involved, how much and how quickly they vaporize etc. It should be possible to measure this current though by measuring its magnetic effects. Any such large current will be quickly stopped by the breakers so the measurement must be very quick.

Answered by Ján Lalinský on December 20, 2020

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