# Help with the Interpretation of the Closed Line Integral of Current Density Resulting from an Electrostatic Field

The line intergral of a static electric field around a closed loop is:

$$\unicode{x222E} \mathbf{E} \cdot d\mathbf{l}=0$$

For an ohmic material, this is equivalent to: $$\unicode{x222E} \frac{1}{\sigma} \mathbf{J} \cdot d\mathbf{l}=0 , \tag{1}$$ where $\sigma$ is the conductivity.

I am trying to understand the meaning of this statement (rephrased) from my textbook regarding this equation:

"A steady current cannot be maintained in the same direction in a closed circuit by an electrostatic field. The energy to create the motion of charge carriers must come from a non-conservative field, since a charge carrier completing a closed circuit in a conservative field neither gains nor loses energy."

Specifically,

1. Why does equation (1) imply that a steady current cannot be maintained in a closed circuit by an electrostatic field?

2. Why does the fact that a charge carrier completing a closed circuit in a conservative field neither gains nor loses energy imply that the energy to create the charge's motion must come from a non-conservative field?

The statements are not correct. The equation $$\unicode{x222E} \mathbf{E} \cdot d\mathbf{l}=0$$ hold in all circuits without induction, i.e. a changing magnetic flux in the loop. It also holds in a circuit where you have a stationary current flow due a battery voltage (EMF) where the electrostatic field is maintained by an electrochemical process. Equation (1) holds only in a stationary situation in resistive materials. Also, the energy in a battery circuit doesn't come from a non-conservative field. It is provided by chemical reactions.
• @TheOblivious - I think that he meant that when an electrostatic field is created by mobile charges, like charges on a capacitor, you cannot maintain an indefinite steady current flow in the circuit because the charge, which flows away causing the current, will eventually be zero. This is the law of charge conservation expressed in the form of the continuity equation $$\nabla \vec J=-\frac {\partial \rho}{\partial t}$$. If you want to maintain a current in a purely resistive circuit, you need an electrical loop field induced by a varying magnetic field, which is not conservative. – freecharly Feb 14 '18 at 18:10