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If amps = volts / ohms, and ohms is 0, then what is x volts / 0 ohms?

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In the world we live in, with the accuracies we can generate it is an observed fact the R=V/I.

Infinities need careful interpretation if they happen in the physical world.

In this form when there is no current one talks of infinite resistance ( seen also on the potentiometers sold) .

When one reverses the equation to the form I=V/R one has to be careful to see if there can be any material where R is 0. There are no such every day materials because they are composed by atoms tied together with electromagnetic forces which will always display some resistance to change of status at normal temperatures.

But there exist special materials under special conditions, superconducting materials and superconductivity, which take advantage of the quantum mechanical behavior of certain metals, and there one achieves practically zero resistance and very high currents indeed, according to the voltage applied.

The simplest method to measure the electrical resistance of a sample of some material is to place it in an electrical circuit in series with a current source I and measure the resulting voltage V across the sample. The resistance of the sample is given by Ohm's law as R = V/I. If the voltage is zero, this means that the resistance is zero.

Superconductors are also able to maintain a current with no applied voltage whatsoever, a property exploited in superconducting electromagnets such as those found in MRI machines. Experiments have demonstrated that currents in superconducting coils can persist for years without any measurable degradation. Experimental evidence points to a current lifetime of at least 100,000 years. Theoretical estimates for the lifetime of a persistent current can exceed the estimated lifetime of the universe, depending on the wire geometry and the temperature.

The current in the superconductors is not found by this simple formula, but theories have been developed and methods of measuring it use the magnetic fields generated.

The LHC uses high power superconducting magnets to achieve the high magnetic fields it needs. The problem is technological, keeping the superconductors cooled and the high power needed under control.

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You can't put x volts over 0 ohms, the material must be at equal potential everywhere.

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It must be equal potential everywhere because there are 0 ohms of resistance? –  Waffle Jul 9 '12 at 3:45
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@Waffle: precisely. –  Ron Maimon Jul 9 '12 at 7:56

In elementary circuit theory, one learns to plot I-V characteristics for circuit elements. The plot for a resistor is a line, passing through the origin, with a slope of 1/R. So, the plot for a zero ohm resistor is simply the vertical axis. Looking at such a plot, you immediately see that the voltage across a zero ohm resistor is zero for any current.

Moreover, if you draw the circuit diagram for your question, you see an ideal voltage source connected across a wire, i.e., an ideal "short circuit". If the voltage source has a voltage $V_S \neq 0$, applying KVL yields a contradiction: $V_S = 0$.

So, as Ron points out and the above makes clear, there's a contradiction in your assumption that there can be a voltage across zero ohms.

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the amperage is current flowing through the superconductor. All superconductor have a critical magnetic field they can counter before the superconducting phase breaks. This critical magnetic field also implies a critical electrical current, because all current will generate an associated trasverse magnetic field.

What happens to the omhic law in a superconductor is that a non-zero (but below critical) current can be sustained indefinitely even in zero voltage, so the ohmic law relationship becomes an indefinite $\frac{0}{0}$ expression, which can be understood as a breakdown of the dependence between the quantities

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