Why isn't this largest voltage source given as negative?

[kostoglotov]

This isn't actually a homework problem, but it is an example in a my textbook that I have a question about...

imgur link: http://i.imgur.com/2OglMH4.png

Ok, I wanted to clarify some things. I understand that current flowing from a positive terminal to a negative terminal represents a positive voltage drop. I can see why the second voltage source (-8V) is a negative, since current is flowing against that voltage gradient within the component.

Two things

(1) the current is also flowing against the voltage gradient inside the 32V source, but the voltage is given as positive...is this because it is the dominant voltage source...so it represents not a voltage drop but a voltage rise, and as such, it is equivalent to a negative voltage drop...so a negative times a negative...?

(2) the resistors are passive elements, surely their polarities should be determined after we've figured out which way the current is flowing, no? Because the polarity of the bottom resistor should be reversed shouldn't it?

 

[BvU]

Hi,

The book exercise is trying to mislead you (it mislead me in first instance). The exercise is in fact trying to teach us a lesson in sign conventions: If it says -8V next to a circle with a + and a - sign, the idea is that the + voltage is 8 V lower than the - voltage. That is not meant to be a conclusion but is meant as a given !

That way you get a total voltage of 24 V over the two resistors and hence a current of 4 A.
In the top resistor the current is from left to right, so the side marked + is at a higher voltage than the side marked - and therefore v₁ is positive.
In the bottom resistor the current is from right to left to, so the side marked - is at a higher voltage than the side marked + and therefore v₂ is negative.

(1) the current is also flowing against the voltage gradient inside the 32V source, but the voltage is given as positive...is this because it is the dominant voltage source...so it represents not a voltage drop but a voltage rise, and as such, it is equivalent to a negative voltage drop...so a negative times a negative...?

(2) the resistors are passive elements, surely their polarities should be determined after we've figured out which way the current is flowing, no? Because the polarity of the bottom resistor should be reversed shouldn't it?

(1) It (going from - side to + side) always represents a voltage rise if the value for v (32 V here) is positive (so also if it is not the dominant voltage source).
(2) resistors have no polarity, but -- as you say correctly -- when you found out the direction of the current, there is a side that is a a higher voltage than the other side.

The + and the - marks simply dictate a sign for v₂. So if you connect a voltmeter (+ on left, - on right) then it shows $-$8 V.

The example isn't trivial, but someone who doesn't understand it does not benefit and only getst more confused. So it's good you check!

A double check would be to repeat the exercise with -44 V on the right instead of - 8 V !