Is the strength of an electric field the same throughout a conductor connected to a battery?

Question

Is an electric field the same all over a conductor?

As electrons move further in a conductor they start coming from high potential to low when moved from high potential end of conductor(cathode) to low potential end of conductor(anode,) so the potential difference between points keeps on decreasing thus causing a decrease in strength of electric field.

If this is the case then the current should be different through out a conductor (different strength of electric field between different points in conductor thus different accelaration.)

Where am I wrong?

The red dot is an electron. The conductor is connected to a10V battery. Section A is the midpoint of the conductor, so the voltage between section A and C should be 5V while between B and C is 10V so according to it the electric field between A and C should be weaker than the field between B and C so accearation should be lower between A and C than B and C, so doesn't it change the current at different points in the conductor as electrons at point B experience a higher accelaration thus more current than electrons at A?

Answer 1

For a conservative electric field, \$E\$ is equal to the (negative of the) gradient of the potential. That is:

$$E = -\nabla V$$

If the electric field is also uniform, then in the direction of the electric field, \$E\$ is equal to the change in potential per unit change in distance.

$$E = -\frac{\Delta V}{\Delta x}$$

If a wire has constant width and uniform resistivity, then the above equation will apply.

So, although the potential will drop along the length of the wire, the electric field will be uniform.

Note that the potential drop along a length of wire is given by

$$V = IR$$

The resistance of normal conductors is "low" and is often treated as negligible. When the resistance is treated as negligible, the voltage drop across the length of wire is negligible as well, hence the oft repeated mantra that the electric field inside a wire is 0. (If the current is non-zero, and the resistance is non-zero, then the electric field will be non-zero. It is just that for many purposes, we can ignore it.)

Answer 2

Current is by definition the flow of electrons. If you had different currents in different parts of a circuit then you would quickly get an excess or a deficit of electrons in one part, and this would change the voltages around the circuit, so the currents would quickly equalise again. In reality this does happen but the voltage differences are usually tiny.