# Safety in ultracapacitor discharge regarding possible effects of induced voltages

### Problem

I am trying to analyze the safety of two experiments involving ultracapacitor short-circuit discharges.

In the first experiment one has a discharge of a $0.45 \,\mathrm{F}$ capacitor with an initial current of a about $I = 10\,\mathrm{kA}$ and an initial voltage of about $U_0 = 60\,\mathrm{V}$.

The second experiment is about a $2600 \, \mathrm{F}$ capacitor with an initial voltage of $U_0 = 2,7\,\mathrm{V}$ and an initial discharge current of about $1\,\mathrm{kA}$.

I know that there are dangers for example by evapored metal or uv-emissions of lighnings etc. For this question I want to discuss if there are any considerable dangers about electric shocks through induced voltages.

How to derive a good criterion to test the safety of those experiments in this particular aspect?

### What I tried so far:

I think it will be possible that large voltages are induced in this setup depending on the switching times with determines the changes of the magnetic flux in this case together with the large currents flowing.

So we may not apply the usual safety criterion of how large currents are flowing continously through the human body, but if the energy deposited to the body is low enough in a short enough time.

To illustrate what I mean with that criterion just consider a demonstration parallel capacitor. I think many physics teacher touched such a device with for example $5 \, \mathrm{kV}$ without any problems. The very high initial currents are not dangerous in this case because they drop fast enough to almost zero and deposit not much energy to the human body. Usually an energy below $E_{s} = 350\,\mathrm{mJ}$ is considered as safe.

However I am not sure if the discharge time should be below a certain value such that this criterion does apply or is it valid independent of the discharge time?

I think it would be a good idea to derive a criterion for a tight upper limit of possibly deposited energy in this setup (modulo the effect of the discharge time).

Consider the examples above recardig the energy stored in the capacitors $E = \frac{1}{2}CU^2$

In the first case:

$$E = 810\,\mathrm{J}$$

In the second case:

$$E = 9477\,\mathrm{J}$$

So the energy contained in those capacitors is much larger than $E_{s}$.

However this is not the energy that will be deposited via induction into the human body since that happens only if the magnetic flux, i.e. in this case the currend changes. This happens when switching the circuit on. For this you have to consider the inductivity $L_{\mathrm{Wire}}$ of the wire.

The highest rate of change of the curent will take place at time $t=0$ with

$$\dot I(0) = \frac{U_0}{L_{\mathrm{Wire}}}$$

which means the highest $\dot \Phi$. This decays exponentially. Say after a time $t_{\mathrm{Switch}}$ (but I don't know how to get this time), the change of magnetic flux per time may considered as safe. Then one could calculate the energy $\Delta E$ dissipated by the capacitor via:

$$\Delta E = E(0) - E(t_{\mathrm{Switch}})$$

Where

$$E(t) = \frac{1}{2}C (U_0 \cdot \exp(-\frac{t}{RC}))^2$$

Only a part of this energy may be deposited in the human body due to an induced electric field (maybe most of is is dissipated via heat in the wire directly). But if $E_s > \Delta E$, I would consider the experiment as perfectly safe regarding the scope of this question.