Consider a circuit where 2 resistors are connected in parallel

[Oerg]

This is a conceptual problem

Consider a circuit where 2 resistors are connected in parallel, which are in turn connected to another resistor in series. Let the resistance of the two resistors in parallel be R_1 and R_2. The resistance of the resistor in series is R_3

Normally we would add the resistance like this to find the total resistance

R_{total}=R_3+\frac{1}{\frac{1}{R_1}+\frac{1}{R_2}}

however, this works out different when i derive the resistance of the circuit anew from kirchhoffs law which would give

R_{total}=\frac{1}{\frac{1}{R_1+R_3}+\frac{1}{R_2+R_3}}

am i missing something? or is it wrong to apply the resistance formula in the case where there is another resistance connected in series?

 

[ehild]

Show your work. How did you get your formula? Actually, the formula for parallel and series resistors are derived from Kirchhoff's laws, you should get the same result.

ehild

 

[Oerg]

well if you simplify the expressions you would not get the same result, it is immediately apparent that there will be a {{R_3}^2} term in the second expression but not in the first.

The second expression was obtained by considering 2 different loops and combining the resulting current, same as you would for the derivation of the formula for the combination of resistors in parallel.

 

[willem2]

Your 2 loops are: +side of voltage source -> R3 -> R1 -> - side of voltage source
and
+side voltage source -> R3 -> R2 -> - side of voltage source ?

both loops contain R3. The voltage across R3 depends on the currents of both loops.
The voltage across R3 is not equal to R3*(current in one of the loops)

If 2 loops share a voltage source, it isn't a problem, because the voltage across it is
always the same.

 

[ehild]

The picture shows your circuit.
According to Kirchhoff''s Current Law, I₃=I₁+I₂.
According to the loop Law,
U_BC+U_CA=U_BA, that is
E-I₃R₃-I₁R₁=0
and E-I₃R₃-I₂R₂=0.
The total current is I₃, the total voltage is E, and the resultant resistance between A and B is
R_AB=E/I₃, and it is the same as the first formula.

ehild

 

[Oerg]

Ahhh, you even have a diagram, I am so touched. Thanks for your effort on the forums.

I understand now, in my loop, I had assumed that the same current run throughs both resistor and that I could get the resultant current by adding up the current from the other loop. This is of course not equivalent.