Stationary charge near a current carrying wire: Experiment to check it at home

Question

From what I have read so far, I arrive to a conclusion that a stationary charge must experience a force when it is near a current carrying wire. A stationary electron should get attracted and a stationary proton should get repelled. I see there are some who agree to this, but there are many who say that a stationary charge won't experience ANY force near a current carrying wire.

I went into youtube to find if anyone has done it experimentally but no luck. There are tons of videos for two parallel current carrying wires. But none for stationary charge(s) near a current carrying wire.

I assumed this must be fairly simple to do.

1. Take a couple of AA batteries
2. Connect them end to end using a striped wire
3. Charge up a Balloon by rubbing it against a woollen cloth.
4. Bring the battery/wire setup near the balloon and it should get attracted if there is a field.

But since no one seem to have tried this, I wanted to ask if there is any flaw in the above experiment? Would it not work to prove or disprove the theory?

Update: Okay, it turns out this is not so simple and cannot be carried at home. But here is what I found:

We present several kinds of experiments. Some map the lines of electric field outside resistive wires carrying steady currents. Others map the equipotential lines outside these conductors. Other experiments directly measure the force between a charge test body and a wire carrying a steady current, when there is no motion between the wire and the test body. Anther experiment measures the charging of an electroscope connected to different points of a circuit carrying a steady current. Yet another experiment describes how to obtain a part of the surface charge in different points of the circuit, showing also how to verify if it is positive or negative and also its magnitude. Bergmann and Schaefer present some experiments in which they mapped the electric field lines [172, pp. 164-167] [173, pp. 197-199]. They comment that due to the great conductivity of metals it is difficult to utilize metals as conductors in these experiments. Metals cannot sustain a great potential difference between their extremities, so that they produce only a very small external electric field. For this reason they utilize graphite paper strips of high resistivity and apply 20 000 to 40 000 volts between their extremities in order to produce a steady current along the strip. They ground the center of the strip to put it at zero potential, so that the lines of the electric field are symmetrically distributed around it. They then spread semolina in castor oil around the strip, and obtained the result shown in Figure 3.1. The central straight dark line is the paper strip carrying a steady current. The particles of semolina polarize due to the external electric field and align themselves with it, analogous to iron filings mapping a magnetic field. It should be observed that along the external surface of the conductor there is a longitudinal component of the electric field. This aspect differentiates it from the electric field outside conductors held at a constant potential (in which case the external electric field in steady state is normal to the conductor at every point of its surface), as has been pointed out by Bergmann and Schaefer.

Answer 1

Yes, there is an electric field outside a current carrying wire, and in absence of external charges, it is mainly due to the surface charge on the wire that is necessary to shape the electric field inside the conducting material.

So, in principle a static charge near a current carrying wire - more precisely near a circuit carrying a stationary current - will be subjected to the electric field of the surface charge on the wire, and also of the interface charge that will develop between components. The space around the circuit will therefore see an electric field generated by surface and interface charge, and a magnetic field generated by the charge flowing inside the conductor.

Mind you, the surface charge distributions depends on the geometry of the circuit and the position of other elements such as resistors (which develop charge at the interfaces with the conductor). Here is one simulation from the paper "A semiquantitative treatment of surface charges in DC circuits" by Rainer Muller (first link on Google)

As you can see, sometimes the electric field impinges on the wires, sometimes it diverges, sometimes is nearly parallel. There is no 'general rule': geometry and boundary conditions can result in complicated configurations. Near the battery and near the resistors the electric field can be much greater than the electric field inside the conductors (where $E=j/\sigma$ is usually very small due to the high conductivity.

If you are interested in the electric field around an infinitely long rectilinear wire you might look up Sommerfeld's Lectures on theoretical physics, volume 3, Electrodynamics, p.125 "Detailed treatment of the field of a straight wire and a coil". Be careful, the solution depends on symmetry and he is showing the midpoint between infinitely far away extremes.
A detailed treatment of the electric field outside a wire is given in the paper "The Electric Field Outside a Stationary Resistive Wire Carrying a Constant Current" by Assis, Rodrigues, and Mania. The authors "show that this force is different from zero and present its main components: the force due to the charges induced in the wire by the test charge and a force proportional to the current in the resistive wire." They also "discuss briefly a component of the force proportional to the square of the current which should exist according to some models and another component due to the acceleration of the conduction electrons in a curved wire carrying a dc current (centripetal acceleration)."
In the case of a linear wire one meter long and 1mm in diameter subjected to a potential difference of 1V, the authors computed the forces on a test charge of 10^-9 C and found that the force due to electrostatic induction is ten thousand times stronger than the force due to surface charge, while the force due to secondary effects is a million times weaker.

The first experiments to demonstrate the presence of an electric field outside the wires is a circuit were documented by Jefimenko in 1962. He used high voltage circuits and thin grass seeds (spread unto a glass plate) that did not introduce any net charge on their own, but instead got polarized and oriented themselves along the electric field lines.

One of Jefimenko's experiments, picture taken from the paper "Surface Charges and External Electric Field in a Toroid Carrying a Steady Current", by J. A. Hernandes and A. K. T. Assis.

Surface charge is very small and these effects are hard to demonstrate, as you realized, but the theory is clear: there is an electric field outside the conductor and charge near it will experience its effects.

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References:

DOI:10.1119/1.1941887.
Oleg Jefimenko.
Demonstration of the electric fields of current-carrying wires.

DOI:10.1119/1.18112.
John D. Jackson.
Surface charges on circuit wires and resistors play three roles.

DOI:10.1119/1.3480576.
Jacobs, Salazar, Nassar.
New experimental method of visualizing the electric field due to surface charges on circuit elements.

Answer 2

Since $\vec F = q\vec v \times \vec B$, you should get $\vec 0$. But experimentally checking theoretical answers is entirely in the spirit of physics.

Whether there are any flaws in your experiment depends on how carefully you carry it out. The most likely problem is that you will have some small force that does not originate from the current. How will you show that the current does or does not cause any force you find? There are already some suggestions in the comments about this.

Keep in mind that experiments like what you propose, and many others, have already been done. The laws relating current to magnetism were explored by Michael Faraday in the early 1800's. Many other scientists following him have contributed. Look some more before you decide it hasn't been tried.