# Why isn't the magnitude of the electric field in a circuit zero?

I've been re-reading my knowledge of circuits and how current flows as well as the surface charge distributions in the steady state, since it's been quite some time and we've just started studying them in first year.

My question is concerned about why the electric field in the circuit is non-zero. Before the steady state, since the wire of a circuit is a conductor and you have an electric field applied to it (by the dipole of the battery), the charges distribute themselves due to the electric force by this electric dipole. I'm curious to know why the electrons continue distributing themselves until there is a non-zero electric field in the wire, why don't they distribute themselves such that the magnitude of the electric field anywhere within the wire (a conductor) is zero? My current thoughts on this could be due to the battery constantly increasing the electric potential of electrons...

• Your current thoughts seem to be in the right direction. – AV23 Apr 21 '15 at 13:24
• I'll give it a good think for a while longer, I'll report back later if I think of a potential solution - Also, pun intended? – charl1e Apr 21 '15 at 13:35

In the dynamic case (current flowing), the redistribution of the electrons takes work to overcome resistance.

As you know, $V = I \cdot R$ and $E = \frac{dV}{dx}$. There needs to be an electric field in order for current to flow - and the current flows because of the potential difference.

You are right that the electrons will try to flow to cancel the field - but in the presence of resistance, they can't.

• I don't think so... if a metal sphere is polarised by an external electric field, the metal sphere still has resistance, yet the net electric field in the interior of the sphere is zero - Why isn't it the same for a circuit? – charl1e Apr 22 '15 at 2:24
• I'm thinking it must be due to the battery causing a constant non-zero electric field. I'm imagining the most simple circuit, basically a big rectangular metal block, with a net positive charge on one end and a net negative charge on the opposing end, initially. Since this metal block is polarised, there's a non-zero net electric field in the block; causing the mobile-electron sea to shift ever so slightly. If I continued adding electrons to one end by 'grabbing them from the positive end' and moving them to the negative end, there'll always be a non-zero electric field in the block... – charl1e Apr 22 '15 at 2:30
• I think this same principle applies to every circuit, just it's much more complicated since complex charge distributions come into play, which confuses me – charl1e Apr 22 '15 at 2:32
• So the mobile electron sea in the circuit is actually trying to continuously cause the net-electric field to be zero, but since the battery is raising their potential over and over again it never reaches that state; I think – charl1e Apr 22 '15 at 2:40
• In the dynamic case you are always doing more work on electrons - not so in the static case. – Floris Apr 22 '15 at 11:58

The way you conclude that there's no field inside a conductor in equilibrium is this: there are no currents in equilibrium, therefore (by Ohm's law) the electric field is null. Out of equilibrium and in the presence of currents, Ohm's law says you get electric fields.

If you want to look into microscopic explanations of Ohm's law, check out the Drude Model. Basically, to have a current you need an electric field that gives energy to the electrons, to compensate the energy they lose to scattering.