# Can charge flow between two points if their potential difference is 0?

Suppose there is a circuit with the conducting wires having zero resistance. The circuit is divided into two branches with one branch having no resistor and the second branch having one resistor with resistance r. In what manner will charges flow through the two branches?

MY REASONING : My book states that a potential difference is required for the flow of charges from one point to another. Since the first branch has no resistance, according to V=IR, the potential difference between the points is zero and hence no charge will flow through the two points and all charges will take the second path.

• The common answer is for two resistors in parallel, where one resistor has value 0. The REST of the circuit, however, may matter: if the wire without resistance is the winding of an AC generator, for example, there's EMF in addition to potential difference involved. – Whit3rd Dec 16 '16 at 8:24
• Related-physics.stackexchange.com/questions/69919/… & physics.stackexchange.com/questions/51875/… Might answer your problem – SNB Feb 4 '18 at 16:06

Can charge flow between two points if their potential difference is 0?

Define current:

Electric current is the rate of charge flow past a given point in an electric circuit, measured in Coulombs/second which is named Amperes. In most DC electric circuits, it can be assumed that the resistance to current flow is a constant so that the current in the circuit is related to voltage and resistance by Ohm's law.

These relations describe electric circuits, i.e. for current to flow continuously there should be a potential difference introduced and an electromotive source which provides the energy. Otherwise the currents are transient.

Like mechanical potential energy, the zero of potential can be chosen at any point, so the difference in voltage is the quantity which is physically meaningful. The difference in voltage measured when moving from point A to point B is equal to the work which would have to be done, per unit charge, against the electric field to move the charge from A to B. When a voltage is generated, it is sometimes called an "electromotive force" or emf.

In this simple circuit we see a voltage difference imposed by a battery on the left. The voltage is the same in the conducting wire ( assuming zero resistance wire) there is a voltage drop across the resistor, and then there is no voltage difference on the return wire. So the answer is , yes, a current can exist in a closed circuit on the the zero resistance wires. It is enough that an electromotive force supplies the energy for the current.

For increments in the circuit where there is no voltage drop Ohms law is undefined, as 0/0, it cannot be used to find the current.

The crucial term is that in a "closed circuit" there must exist a voltage drop, otherwise the problem is undefined. In an open circuit there is not current

I am assuming your book refers to a parellel circuit of a resistor of resistance R and a simple wire with zero resistance. If we do not take into account thermal energy, we could move charge my applying a potential difference at the two ends of the parallel. This is a limit case of current divider. The formula for the current in a wire placed in parallel is Ix = (Rt*It)/Rx, where Rt is the equivalent resistance of the parallel, It is the total current that enters at the "beginning" of the parallel, and Rx is the resistance of the branch. In the case of a branch that has 0 resistance, Rx would be 0. By taking the limit as Rx approaches 0 of that formula (cause you cannot just divide by zero), we obtain that the current in the branch with zero resistance approaches infinity. That happens because current always divides in such a way that will minimize the energy expended. If some current had to pass across the resistance, it would be dissipated as thermal energy, so it won't pass that way. In fact, the current encounters a path that is completely free of resistance, so it will "want" go that way! That is a short-circuit. Now, an interesting question would be to ask if charge can move in a wire that is not connected to any battery or device that could set a potential difference across its ends. The answer is YES, if we consider a wire which temperature is above absolute zero. In fact, temperature is the manifestation of atoms movement inside a material. If we consider a transverse section of a wire and count the electrons (charges) that will pass through it in time (which is the definition of current), we'll find that the average is zero (since we have not applied any potential difference), but that is only because there will be an equal number of charges going "to the left" and "to the right"! So, thecnically, charges can move with no potential difference too.

• Underrated answer, maybe Mathjax would help – SNB Feb 4 '18 at 16:10

I don't think so. There must be some potential difference for a charge to flow from one point to another. To move a charge from one point to another there is need of doing work on charge. This work done is provided by the potential energy stored in charge due to applied potential difference.

This question is difficult to answer without a circuit diagram so i will use an assembly of circuit elements that I think will answer your question. Take the case of two terminals (a and b). a and b are connected by two paths; one path goes through a resistor, one does not. With no voltage difference between a and b, you argue correctly that no current would flow. If a positive valued voltage V (e.g. 9V) were applied from a to b, current would flow from a to b though the wire with no resistor. This wire is called a "short circuit" because it diverts functionally all the current from the resistor path to its own path. Short circuits are able to discharge batteries and other electrical potential storage devices very quickly. So, if the 9V volts across the terminals were due to a battery, the terminals would very quickly reach a voltage equilibrium and current would cease to flow from a to b. If the voltage across a and b came from a wall outlet, barring safety devices like fuses, current would continuously flow in the zero resistance wire from a to b.

In summary: if a voltage source is applied across a and b, the terminals a and b are not equipotential surfaces until the voltage source is drained. Given the "choice" of a lower or higher resistance path, more current will flow through the less resistive path. In the extreme case of a short circuit, all current will flow through the short. Notice that solving for I in this case requires dividing by zero: I=V/R as R approaches 0. And finally, if there is no voltage difference between a and b, no current will flow.

$V=IR$ and its AC partner, $V=IZ$ are strictly related to circuits with impeding elements, and are not general principles. There are ways to produce currents which persist without the continual replenishment of energy via some chemical, mechanical, or secondary electronic means, namely, magnetic induction in superconducting coils.

Currents are induced in close superconductors through the use of an oscillating magnetic field. When the field is removed, the current persists indefinitely until energy is sucked out by having that current itself induce a current in an outside coil. This technique is used in high precision NMR units.

Charge will flow through the least resistive part since the circuit possess one least resistance wire therefore the whole current will pass through that wire . Current through resistance wire will be 0