Single Loop Potential difference -- sign issue

[simphysics]

Homework Statement

Find Va-Vb in the following circuit.
R=3 OHM
Ɛ1=10V
Ɛ2=4V

Relevant Equations

Va-Vb=Ɛ1
Va-Vb=iR-Ɛ2

Hello everyone, I've been struggling with this problem for 1 hour so far.

Here's how I've tried to solve it. as i supposed, the current flows counterclockwise and i=2A.

it doesn't matter which path i take, cause i should always arrive at the same potential but i actually don't and i can't get why.

If taking Vab clockwise:

Va-Vb=Ɛ1 and i assume it positive since the current flows through it from the negative pole to the positive one. so Va-Vb=10V

If taking Vab counterclockwise

Va-Vb=iR-Ɛ2=6V-4V=2V

I'm assuming Ɛ2 negative since the current flows through it from the positive pole to the negative pole.

The values are different and I can't get why.

Thank you! :)

homework-and-exerciseselectric-cir

 

[Orodruin]

Current direction is irrelevant for the emfs. All that matters is the polarity. You therefore has the wrong sign for the second emf.

 

[simphysics]

Got it, so what's the rule for the polarity?Thank you!

 

[Orodruin]

If you go from + to -, potential drops. If you go from - to +, potential increases.

 

[simphysics]

sorry but i still can't get over it...
when i calculate Vab counterclokwise i go from + to -, so why should i take it positive?
thank you

 

[Orodruin]

Why did you take the other emf positive?

 

[simphysics]

well...i don't know at this point

 

[Orodruin]

Which direction you go and how you count is conventional. Computing Vab = Va - Vb you can either go from a to b and count potential drops or go from b to a and count potential gains. Since you say you want to go ccw from a to b, you should be counting potential drops. The drop across the resistor is iR because you are going in the direction of the current. The drop across the emf is E2 because going from + to - you get a potential drop.

Same thing happens when you take the other way across the other emf. You get E1 because you get a drop of E1 going from + to -.

 

[simphysics]

everything's clear now, thank you so much!