So my question is like the figure. It is given that V0 = 2V and asked what Is the independent current is.

I wrote i1, i2, i3 and i4. I took the bottom essential node as ground zero.

Yellow node is 2V, therefore 2V = i4*4, i4 = 0.5A.

Vblue-2V=8*0.5, Vblue=6V, so i3 = 2A.

i2=i3+i4, which means i2 = 2.5 A

Vgreen-6V = 2.5*2, Vgreen = 11V = (12+10)*i1, i1 = 0.5 A

Is = i1 + i2 = 3 A.

Now my questions.

1. Did I solve it correctly? I'm not quite sure about some steps.
2. Does the independent current source has a voltage on its own? Because I tried to solve this question with nodal analysis and I couldn't decide what to do with Is.
3. It was just an instict not to write any current flowing to the node a or node b. Is it always the case in the open circuits?
4. Is it an easy question? At first, I couldn't find the appropriate position to start. I said current flowing through 4 ohms is i1 and then I had to call the current through 8 ohms i2, which turned out to be wrong in my final solution. How do we decide the best way of determining the unknown currents?

Thank you for any answer in advance. I'm not a native speaker, so if you have trouble understanding anything I've written, I'd gladly explain what I really meant.

 

AD1. Your answer look good.

AD2 The voltage across the current source is equal 11V.

AD3 Yes if we have a open circuit the current is always equal 0A.

AD4 The only thing you need is experience in solving those type of circuit.