Basic circuit - connected parallel branches

[micah20]

Note: this isn't an actual problem from a textbook; it is something that I've been wondering about because I was never taught how to do this in school or in my physics textbook.

Homework Statement

In this very basic circuit, how do you determine the direction and value of current through each resistor and through the vertical connection between the two parallel branches?

The Attempt at a Solution

If it weren't for the connection between the two branches, you would combine each series pair of resistors, and then use I=V/R to determine the current through each branch. But when the parallel branches are connected like this, how do you determine current direction and value? Thanks.

 

[The Electrician]

Suppose there were a resistor connecting the middle of the two branches instead of just a (zero ohm) wire? Would you know how to solve the circuit then?

I ask because there is more than one way to solve this problem, and the choice of a method might depend on your level of knowledge of network solution techniques.

 

[n.karthick]

If it weren't for the connection between the two branches, you would combine each series pair of resistors, and then use I=V/R to determine the current through each branch.

Hmmm. If you refer to the figure, you can find two pair of resistors in parallel are in series, and each pair can be reduced to single resistance, so that you will be left with two resistors in series. Now you can simply add these resistances and find the total current.
Hope you know how to find current through individual resistors.

 

[micah20]

Okay - I think you mean it can be simplified like this:

1) separate it into two parallel combinations in series with each other (top)

2) simplify each parallel combination into single approximate values using the formula 1/Rparallel = 1/R1 + 1/R2 , which comes to approx. 180 ohms each (middle)

3) add the two series resistances together to get a single effective resistance of the circuit of approximately 360 ohms (bottom)

Is this correct so far? Electrician - I'll answer your question but first I need to know if I'm on the right track so far.

 

[n.karthick]

Yeah, you are right with your answers. Once you know the total current flowing, you can find currents through individual resistors in parallel.

 

[The Electrician]

You asked:

"In this very basic circuit, how do you determine the direction and value of current through each resistor and through the vertical connection between the two parallel branches?"

What you've done so far, while ok as far as it goes (you haven't calculated the current through each resistor), provides no answer to the part of your question in red (the hard part).

Do you still want that answer? It's this part that requires some additional skill at circuit analysis.

 

[micah20]

Okay, the total current = V/Rtotal = 12/360 = 0.0333A or 33.3mA. To find the current through each resistor you must first find the voltage drop across each of the two parallel combinations using deltaV = IR. So the voltage drop across both parallel combinations rounds to approx 6V each (if I had used more decimal places or smaller units I could've been more precise). And each 180ohm resistor passes almost the full 0.0333A while the other two pass very little (again, rounded off). I think this is correct - if not let me know.

Electrician, about what would happen if there were a resistor instead of just a wire connecting the two parallel branches; the attached pic shows what I think you are asking, and I arbitrarily chose a value of 1000ohms for that resistor. I've never been taught how to do this but I'll take an educated guess as to how you would find the direction and value of the current through each part of the circuit. I don't know if/how you can simplify the circuit into two parallel combinations in series with each other like I did when there was just a wire in between. So I'm wondering if you can instead analyze by isolating this circuit into two current paths (shown in red and blue), finding the current flowing through both paths individually, then superimposing both, and where the red and blue currents overlap, the current would be the sum of the two (if flowing in the same direction), or the absolute value of the difference and in the direction of the stronger current (if flowing in opposite directions). Would this be one way to solve? Thanks.

 

[The Electrician]

Okay, the total current = V/Rtotal = 12/360 = 0.0333A or 33.3mA. To find the current through each resistor you must first find the voltage drop across each of the two parallel combinations using deltaV = IR. So the voltage drop across both parallel combinations rounds to approx 6V each (if I had used more decimal places or smaller units I could've been more precise). And each 180ohm resistor passes almost the full 0.0333A while the other two pass very little (again, rounded off). I think this is correct - if not let me know.

This is correct.

Electrician, about what would happen if there were a resistor instead of just a wire connecting the two parallel branches; the attached pic shows what I think you are asking, and I arbitrarily chose a value of 1000ohms for that resistor. I've never been taught how to do this but I'll take an educated guess as to how you would find the direction and value of the current through each part of the circuit. I don't know if/how you can simplify the circuit into two parallel combinations in series with each other like I did when there was just a wire in between. So I'm wondering if you can instead analyze by isolating this circuit into two current paths (shown in red and blue), finding the current flowing through both paths individually, then superimposing both, and where the red and blue currents overlap, the current would be the sum of the two (if flowing in the same direction), or the absolute value of the difference and in the direction of the stronger current (if flowing in opposite directions). Would this be one way to solve? Thanks.

I suppose this would work if you could actually do the part I've colored red above. In the top circuit in the image, when you try to calculate the currents in the red path, how would you deal with the effect of the 10k resistor?

Here's a method to solve the problem with the 1000 ohm resistor, or with just a wire in that position:

http://kauko.hallikainen.org/rw/theory/theory6.html

Does that answer your original question?