Help understanding phasor solution

[doktorwho]

Homework Statement

From the given circuit and provided information draw a phasor and calculate the inductance.

Homework Equations

3. The Attempt at a Solution

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The paper was given to me at class as a solution to our homework and i can't understand how they drew the phasor and drew the conclusion out of it.
I see that they took $U_{12}$ as their phasor axis since its the same for both sides but why do they draw the $I_c$ the way they did? How did they know that it goes up and how did they get the $I_c/2$ part?
The continuation of the left side of the third equation i do get but not the right side. Why is the right side like that?

 

[NascentOxygen]

If the inductor were ideal (with R=0) then I_C would lead I by 180°. If the inductance were zero, making it a simple parallel RC arrangement, then I_C would lead I by 90°. So with both R and L then I_C will lead I by some angle between those 2 extremes. That constrains the shape of triangle that you draw.

You also need to bring into the geometry the condition stated in your data of 2 currents having equal magnitude, viz., I = I_g

 

[Numbskull]

If the inductor were ideal (with R=0) then I_C would lead I by 180°. _g

Do you mean 90° ?

 

[cnh1995]

Do you mean 90° ?

No. I is the inductor current. I_c would lead U₁₂ by 90° and if the inductor were ideal, inductor current I would lag behind U₁₂ by 90°. This means I_c would lead I by 180°.

 

[gneill]

I believe that there is a flaw in the question as posed. Perhaps the angular frequency given is to high? If it is truly $10^7 rad/sec$ then the capacitor reactance is just 10 Ω. The real resistance of 100 Ω in the other branch will make it impossible for it to conduct a current with the same magnitude as the source current, even taking into account resonance effects. If I'm not mistaken the maximum magnitude for the current $I$ would be just 1 mA.