How do I calculate voltage drops and currents in a resistor circuit problem?

[subzero0137]

Relevant equations:
V=IR

Attempt at solution:

When the switches are open, R1, R2 and R3 are in series so the current in that case would be 12 V/ (2 ohms + 3 ohms + 3 ohms) = 1.5 A. When the switches are closed, the current across R3 is still 1.5 A, so the voltage drop across R3 is 1.5 A * 3 ohms = 4.5 V. I don't know how to calculate the voltage drops across other resistors and the currents through them. How do I proceed? I know we can ignore R2 when the switches are closed since it gets shorted, and I know R and R3 are in parallel when the switches are closed.

 

[cnh1995]

Your reasoning so far is correct.

so the voltage drop across R3 is 1.5 A * 3 ohms = 4.5 V.

Yes.
So, what is the voltage across R1? Apply KVL to the loop containing ε, R3 and R1.

 

[subzero0137]

Your reasoning so far is correct.

Yes.
So, what is the voltage across R1? Apply KVL to the loop containing ε, R3 and R1.

ε - (I*R1) - (1.5A *R3) = 12 V - (I*R1) - (1.5*3) = 0 therefore I*R1 = 7.5 V. The current across R1 is 7.5/3 = 2.5 A, therefore the current across R = 1 A. Since the voltage drop across R and R3 is the same (4.5 V), the resistance of R is 4.5/1 = 4.5 ohms. Is that correct?

 

[ehild]

ε - (I*R1) - (1.5A *R3) = 12 V - (I*R1) - (1.5*3) = 0 therefore I*R1 = 7.5 V. The current across R1 is 7.5/3 = 2.5 A, therefore the current across R = 1 A. Since the voltage drop across R and R3 is the same (4.5 V), the resistance of R is 4.5/1 = 4.5 ohms. Is that correct?

It is perfect!

 

[epenguin]

Complicated simplification?
If we call the unknown resistance R, R₄, then does everybody agree that this problem gives in general

R₄/R₁ = R₃/R₂
as the condition for currents in R₃ equal for both cases?
A formula (recalling a bridge balancing condition) that seems to be 'suggestive' - but I don't know what it suggests. Can the circuits be switched around somehow to make it obvious what the answer should be? Is there some known principle involved? Duality or something?