If V=I*R according to Ohm's Law, that implies that dV/dx = I x rho/A across an infinitesimal length of conductor, which is constant at all points along the conductor since the current, resistivity, and area of the conductor are all constant. dV/dx is also equal to the electric field, which I wouldn't think would be constant at all points along the conductor since E is a function of the distance from a source (e.g., the battery that the conductor is connected to), so I would think that E would change depending on the location of the conductor. This change might be negligible for a short length of conductor, but over long lengths, I would expect it to be significant. So why is the voltage gradient constant if the electric field is not? Or is the electric field somehow constant at all points in the conductor?

If \$V=I\cdot R\$ according to Ohm's Law, that implies that \$\frac{\mathrm dV}{ \mathrm dx} = \frac{I \rho}A\$ across an infinitesimal length of conductor, which is constant at all points along the conductor since the current, resistivity, and area of the conductor are all constant.

\$\frac{\mathrm dV}{ \mathrm dx}\$ is also equal to the electric field, which I wouldn't think would be constant at all points along the conductor since E is a function of the distance from a source (e.g., the battery that the conductor is connected to), so I would think that E would change depending on the location of the conductor. This change might be negligible for a short length of conductor, but over long lengths, I would expect it to be significant.

• So why is the voltage gradient constant if the electric field is not?
• Or is the electric field somehow constant at all points in the conductor?