# From voltage to electric/magnetic fields

Hello,

I have electrophysiological data from an array of electrodes. I know the locations of the recording tips of the electrodes. Starting with voltage, I'd like to estimate the electric/magnetic fields around the electrodes.

I'm unsure when it's okay to approximate E=-V/d. That is going from time-series of voltages from positions in space I don't know how to tell if curl(E)=0.

Is there a rule of thumb for when electrostatics equations are okay? How do I calculate curl(E) from data like mine?

Thanks so much.

Without knowing your project, I believe E is defined as V/m, so I agree with V/d.

In electrostatic where E is constant, there would be no time varying B.

$$\nabla X \vec E = - \frac{\partial \vec B}{\partial t}$$

$$\frac{\partial \vec B}{\partial t}=0 \;\Rightarrow\; \nabla X \vec E \;=\;0$$

As I said, I don't know anything about your project, are you sure the voltage is constant ( DC )?

Thanks yungman. And yeah, that's about where I got to. The thing is that I'm starting from voltage. So, to see if curl(E)=0 I first need to calculate E, but the only way I know to calculate E is by assuming curl(E)=0 and using the approximation E=V/d. Problem.

Just to reiterate, what I have are V(t,x) where t=time and x=position in 3d space. From there I'd like to find the electric and magnetic fields.

And I'm not entirely sure what it means for the voltage to be DC. Basically I have the voltage difference between each electrode and a neighbouring reference electrode.

Saying there's a DC voltage just means that the voltage does not change sign with respect to its reference point, e.g. typically your ground.

If you truly have V = V(t,x), then your electric field and, by Ampere's Law, your magnetic field will have time dependence. So for a general V = V(t,x), curl(E) is not 0. If, however, your potentials are slowly varying and DC, you can often make the approximation that the time variation of the electric field is negligible and all currents are essentially static. Under these conditions/approximations, your magnetic field will not have a substantial time component and curl(E)=0. Keep in mind that these approximations still don't allow you to make the further approximation that E=V/d. That approximation is a geometric one so we would need to know more about the set up of your electrodes, but to point you in a direction, it requires that the characteristic dimensions of your electrodes are larger than the typical distance between them (capacitor approximation). If that is not the case, the fields are the solutions to the pde wave equations and there's not much of a way to get around that.

Note also that if you have any charge distribution or material that can support surface charges between your electrodes, E=V/d will more generally not be true.

For static fields, Electric field is the (negative )gradient of potential. (regardless of charge configuration) When you say E=V/d you are approximating the gradient with a difference quotient and that would be the major error source I would think.(assuming there are no spurious electrochemical voltages from the electrodes, but I am no expert on that).
Now if the fields are time varying, then one adds the (negative)time derivative of the (magnetic) vector potential to get the field, but that source of error seems negligible here:

Take a 50mV/ms voltage pulse that one may find in a nerve cell. The associated magnetizing field would be (using free space parameters as an approximation and assuming B and V vary over same length L):

B~ L $${\mu \epsilon}$$ (L-1)50 mv/ms~10-17 Tesla. That is obviously too small to bother with.

However, physiological tissue is conductive;there will be currents associated with the potential differences and those currents these might produce magnetic fields.
Internal tissue may have specific conductivities approaching 1 Siemens/meter. Using 100 mV potential differences around a 1 cm cube varying over 50 ms. I estimate a conductive current contribution to the field strength: (dA/dt~dJ/dt *vol/distance)

dA/dt~(1/50 ms) $$\mu$$ (10A/m^2) cm^3/1cm~2 10-8 V/m.
Again seems negligible.
To estimate the error caused by the difference quotient approximation, you might check self consistency with spread sheet of the voltages and calculated electric fields, create an interpolation function to differentiate.

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