# If gradient of potential is zero, how is there a field?

1. Feb 9, 2016

### Electric to be

Consider a common circuit with some resistors in series. The nodes should have approximately the same potential. I know that truthfully the wire just has small resistance compared to resistors. However, even though the gradient of potential is approximately zero in a node, the same current flows through any node of a circuit as a resistor.

If the gradient of potential here is zero, and is non zero across a resistor, that would mean that the electric fields are different.

However, shouldn't the same amount of current be flowing across all parts of the circuit, and current is proportional to Electric field strength.

Thank you.

As a side question, how does it come to be that the electric field is uniform everywhere in a circuit anyways? Is it because initially it isn't, and as a result of charge buildup throughout the circuit, the circuit compensates to become equal?

2. Feb 9, 2016

3. Feb 9, 2016

### Electric to be

That somewhat helps with explaining why electric field remains constant (though I need to do some more looking into), but my main question is how can current be constant throughout a circuit with a resistor. If current is proportional to E, and therefore voltage, and if there is such a steep voltage drop over a resistor but little to none in the wire.

4. Feb 9, 2016

### Electric to be

Actually, I guess the field doesn't have to be uniform, since the resistivity of a material can simply be larger. I guess my last question is why exactly does current have to be equal throughout? Is this simply to prevent charge buildup? And so what if charge did build up anyways?

5. Feb 9, 2016

### cnh1995

Velocity of electrons is proportional to the electric field. It is like a water pipe with variable diameter. In larger diameter part, speed of water will be less than that in the smaller diameter part. But the "flow" rate (Volume/second) is constant everywhere. Can you extend this logic to resistors and wires?

Last edited: Feb 9, 2016
6. Feb 9, 2016

### cnh1995

Yes. Charge builds up only when it is "extra". Same current implies same charge crosses every part of the series circuit at the same time. Hence, there is no excess charge in any part. Surface charge build-up stops when this stable state is reached.

7. Feb 9, 2016

### sophiecentaur

You can answer this question, using your calculator and putting in the values that are easy to find. The force between two charges q1 and q2, separated by distance x is given by
F= q1 q2/(4π ε0 x2) ( The Coulomb Force)
This will tell you the force you would need to 'squeeze' / build up two charges (say +1C each, corresponding to one Amp for one second, flowing in each direction down a wire and ' piling up' with a separation of say 1cm.
ε0 is the permittivity of free space - look it up and work out the answer. The value of the force should convince you why it can't happen.

8. Feb 9, 2016

### cnh1995

Here, by build-up I mean formation of the surface charge rings. Excess charge ends up on the surface of the conductor.