Electric Field distribution around two-conductor cable

What is electric field distribution around two conductor cable connected to DC power supply? Assume power supply is isolated and not grounded. Assume cable is straight.

Case 1: no current runs through the cable.
One wire is positive, another ? negative. Negative or zero? If we think in terms of charge: positive wire has positive charge in it, electric field distribution is radially outwards. What charge is in negative wire? This will determine total field distribution. If there is no charge in negative wire total field distribution will be that from positive wire only - radially outward. If there are negative charges in negative wire the total distribution form the cable will be like from dipole ? on positive wire side field will be outwards, on negative wire side field will be inwards.
What field distribution is correct?

Case 2: load is attached to cable and current flows. Will the Electric field distribution change?

Addressing Case 1: Wiring is not too different from odd-shaped, odd-value capacitor plates. Everything stays charged to the supply node which it is connected to, either positive or negative.

I suppose I can touch a 12v battery after scuffing my shoes across the rug in cold dry weather. A spark might go to the battery and charge it to net +500 V. The positive pole could be 512v and the negative pole 500v. Normally I would never know this happened. I'm not sure if this bears realistically on your question. It's just a way to look at it from a different angle.

I am thinking about example you provided. Does it mean that distribution and density of field lines in space will not change when we add +500V of charge? But if we take isolated charge field will be proportional to amount of charge? ... confusing. It is easier when things oscillate and return to the same state with known frequency. : )

That depends on where you are measuring from.
If you are between the two conductors, there will be a voltage gradient, and a resulting dc magnetic field.

If you are outside the conductors, or the conductors are very closely spaced, both magnetic and voltage field will cancel.

Its a near field / far field effect.
Where near and far are relative to conductor spacing.

Hi,

You say the secondary is isolated from earth potential.
Then the voltage from earth to the wires is not defined.
In an ideal system you may expect -1/2 of DC voltage at the negative wire, and +1/2 of DC voltage at the positive wire. You won't be able to measure it.
The only thing you know for sure is that there is 12DC between the wires.
In a real system there wil be some AC from mains voltage referenced to earth...but that's not the question.

With current..
The voltage and thus the electric field won't change.
Only the magnetic field.
Close to the wires there theoreticaly is a magnetic field. But DC magnetic field in a limited distance is hard to measure.
In some distance the field of both wires will compensate to zero.

Klaus

I did some modeling - for practical cable filed does not exactly cancel. Picture below is result for wires 3mm dia, 1mm apart, voltage 400V (+200v/-200V). No current. The outer boundary shown: 100 mm dia. Solid red corresponds to E-feild > 1000V/m.
It is nor pretty I agree: field does drop compare to single wire; there are two 'dips' in mid-plane between wires, direction of field vector changes around the circle. Still if I have sensitivity few hundred V/m I can detect voltage in cable from few cm distance. (unless it is twisted pair - this will kill the field :)