What would happen if an ideal voltage source was connected to an ideal wire?

Question

I've seen it mentioned in discussion that an ideal wire doesn't exist (all wires have resistance), but even if it did, an ideal voltage source doesn't exist (all batteries have internal resistance); but what if both existed and they were connected together in a circuit?

Answer 1

but what if both existed and they were connected together in a circuit?

This is entirely academic since an ideal voltage source can source or sink unlimited power for unlimited time and an ideal wire has 0V across for any current.

Thus it shouldn't be surprising that the 'answer' is not well defined:

• there is 'infinite' current which allows for the possibility that an ideal wire can have a non-zero voltage across. In this case, the ideal voltage source is delivering infinite power which the ideal wire dissipates.

Clearly, this is absurd but this shouldn't be surprising since, in fact, the equation this ideal circuit yields is (assuming a 1V voltage source)

$$1V = 0\Omega \cdot I_S $$

which has no solution for any finite \$I_S\$ which is, in fact, the most correct answer in my opinion.

The fact is, there are rules for connecting ideal circuit elements to avoid such contradictory mathematical expressions including the rule that one does not connect two ideal voltage sources in parallel.

Yes, that rule covers this case since an ideal wire is identical to an ideal voltage source with voltage equal to zero.

Surprised? But that's how an ideal voltage source is defined - an ideal voltage source is that two terminal circuit element with a given voltage across for any current through.

Set the given voltage to zero and you have an ideal wire. In fact, this is why, in superposition, when voltage sources are zeroed they are replaced by wires.

Answer 2

Since current flowing in the wire creates a magentic field and subjecting a wire to a changing magnetic field creates an opposing voltage, you will get a continuously incresing current flowing through the wire (the wire will behave like an inductor). And it will eventually become the world's most powerful magnet. Incidentally, this is how superconducting magnets work. They have a very large inductance and hence take a very long time to ramp up and ramp down. And the results of heating up before ramping all the way down can be catastrophic (see: the explosion in the LHC the first time they tried to turn it on due to a bad joint between two of the magnets).

In a superconductor, eventually you will reach a point where the material will no longer superconduct. This is called the critical current. If you exceed this current, the resistance will increase dramatically. This also happens when the magnetic field reaches a certain point, called the critical field. In your 'ideal wire,' you would not see this as the assumption is that the resistance is zero. I'm really not sure what would happen for extremely high currents and high magnetic fields - any physics experts out there who might have some ideas?

Answer 3

The current would begin to ramp up linearly from zero amperes at a rate dependent on then inductance in the circuit (even a straight wire with zero resistance has inductance).

The current would become bigger and bigger the longer you left the source attached, but it would never become undefined. Increasing amounts of energy would be stored in the inductance, proportional to the square of the current.

If you applied a 100V perfect source across a 1uH (very small) inductance perfect wire, in about 10 years the energy stored would be equivalent to the total daily energy output of the sun.

Answer 4

An ideal battery always maintains its voltage no matter what. An ideal wire has no resistance, inductance, or capacitance.

But I=V/R, so the instant you closed the switch it would cause a Divide by Zero. Depending on how this error is handled, one of the following may occur:-

1. The Universe crashes and has to be rebooted, but we don't notice because we are part of the simulation!

2. The calculation is cancelled and nothing appears to happen. Scientists then invent a new field of physics to 'explain' the anomaly.

3. Out of nowhere a disembodied voice says "I'm sorry, Dave. I'm afraid I can't do that."

Answer 5

Anyone who reaches for Ohm's law and Kirchoff's laws and saying voltage/zero = infinite current is in error! Those are low-frequency approximations. Things can happen only at the universal speed constant (speed-of-light). (It will be assumed that the original poster has not called 1-800-GOD-PHYSICS-REQUEST to have the speed of light limit removed.)

What would happen when an thick all-temperature superconducter wire is attached to a one-trillionth ohm internal resistance battery?

The wire has to be at some potential value. Maybe halfway between the potential values of the two battery terminals. Where the wire touches the (-) terminal, electrons in their random fuzzy wavefunction wanderings fall into the wire. More and more. As they move into the wire an electric potential builds, an electric field appears in addition to what was naturally around the battery.

At the other end, electrons fall out of the wire into the battery (-) terminal.

The leading edge of the crowd of electrons at the (-) end can propagate only at the speed of light at best. Same for the leading edge of the dearth of electrons at the other end. Moving regions of charge = moving electric fields = magnetic fields. Changing electric and magnetic fields = radiation.

For an "ideal" zero resistance wire, how do those regions of charge move, and how do new charge carriers enter in? If it runs away to infinity, then also an infinite about of radiation can be expected.

What happens when the crowd of extra electrons meets the dearth of electrons near the middle of the wire? Then, for regular devices, a steady current would be established. Where the cloud of extras meet the cloud of dearth, they cancel out. We have a growing region of normal electron density. More changes to a moving electric field, affecting radiation.

Soon that normal region reaches the battery terminals. Then - we have the original situation where we just touched the wires to the battery! Have we made some kinda oscillator? I doubt it. Electrons are quantum particles, and randomness plays a role, so the leading edges of the various regions aren't sharply defined but fuzzy and maybe get fuzzier. A proper treatment would require using equations for semi-classical electron transport (found in, for example, Ashcroft & Mermin) except, being laws of physics, they'd be hard to apply to a nonphysical system.

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Another way to look at this is by inductance.

In this absurd example, the current is destined to reach a steady state of infinity, the magnetic fields are going to be, uh, rather strong. The inductance of the wire is something to think about. Inductance exists anywhere charge moves through space. "Ideal" wire isn't going to be lacking inductance. Current always takes time to get flowing, to reach its steady state value.

I'm not totally sure, but I suspect that the rapidly increasing magnetic field created by the increasing current offers resistance to the current. Remember, inductors 'like' to maintain steady current flow. The runaway current will increase linearly. The inductor equation
\$V_L(t) = L\frac{di(t)}{dt}\$ is easily integrated to give \$i(t)=\frac{ V \over L}{t}\$ where \$V_L(t)\$ has been set to the battery voltage \$V\$.

At some point, the magnetic field will be strong enough to twist Earth out of whack and we'll finally have winter in San Diego :) I like winter, so I'll fund your experiment...

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Skin Effect

Another thing to think about is the skin effect. Currents like to squash themselves out to the surface of the conductor they flow through. At some point, the current will be so great and so squashed, that even in a superconductor or ideal conductor, the graininess of matter will put a stop to the runaway growth.