## Kirchoff's circuits and the Electric Field

1. The problem statement, all variables and given/known data

I have drawn the direction of the electric field in the picture.

I saw this on a video on youtube where this guy solves circuit problems solely on looking at the direction of the electric field. Basically he follows the current and the electric field

$$-\varepsilon_1 + IR_1 + \varepsilon_2 + IR_2 = 0$$

What is the different theory behind the approach? Is it a coincidence that they both will give the same answer or is one of them wrong? For instance between $$b$$ and $$c$$, the electric field and the current is in the same direction so we have

$$\int_{b}^{c} \mathbf{E}\cdot d\mathbf{s} =\int_{b}^{c} Eds = -\Delta V$$

Which means the potential should be minus, but "according to Ohm's Law, the electric field and the current are in the same direction, so we get +IR" and in the battery we go "against the electric field, so we get $$-\varepsilon$$.

The circuit the guy on youtube (http://www.youtube.com/watch?v=UpO6t00bPb8) does involves an inductor, but I thought I could apply the same principle to regular resistor circuits. Does the equation $$\int_{b}^{c} \mathbf{E}\cdot d\mathbf{s} = -\Delta V$$ no longer hold? Note that the integral isn't a closed loop.

 No I am using $$\vec{E}$$ in my picture as the electric field. Also, if you go to this site http://ocw.mit.edu/courses/physics/8...lecture-notes/ and open the file "Non-conservative Fields - Do Not Trust Your Intuition". On page 2/3 in the pdf, you see him does the same thing again. For the left loop he has $$+I_1 R_i$$ even though he assumed the direction of $$I_1$$ is clockwise and he traverses clockwise