Calculating current in circuit with multiple emf's using Kirchhoff's

[Spudnik]

Homework Statement

Find I_1, I_2 and I_3 if the switch, k, is closed using Kirchhoff's Rules.

Homework Equations

V=IR

The Attempt at a Solution

I've had multiple attempts at this and this one is the one that came closest to a solution.

Setting up 3 equations:
1) I_1 = 1_2 + I_3 => I_1 - I_2 - I_3 =0
2) 12V + I_3*(4 Ohm) - I_1*(7 Ohm) = 0
3) 12V - 9V - I_2*(5 Ohm) - I_1*(7 Ohm) = 0

2) 12 = 7*I_1 - 4*I_3 => 12 = 7*(I_2 + I_3) - 4*I_3 => 12 = 7*I_2 + 3*I_3
3) 3 = 7*I_1 + 5*I_2 => 3 = 7*(I_2 + I_3) + 5*I_2 => 3 = 12*I_2 + 7*I_3

Getting rid of I_3:

84 = 49*I_2 + 21*I_3
09 = 36*I_2 + 21*I_3
---------------------------
75 = 13*I_2

I_2 = 5.75

By substituting in, I get

I_1 = -3.7
I_3 = -9.5

Is this anywhere near correct, I don't know why, but somehow this seems fishy to me :S

 

[collinsmark]

1) I_1 = 1_2 + I_3 => I_1 - I_2 - I_3 =0

I think your first equation is not correct. Pay close attention to the direction of the arrows. See how the sum together at either center top node, or center bottom node. You'll find that

I₂ = I₁ + I₃

(not, I₁ = I₂ + I₃ that you originally had.)

[Edit: Oh, and Spudnik, welcome to Physics Forums!]

 

[Spudnik]

OK, I redid it with the new formula and I got

I₁=0.25
I₂=0.25
I₃=-2.5625

Clearly this does not fit the initial equation I₂=I₁+I₃

It's strange though because those are the same values I got for I₁ and I₂ for part A) where it asked for the currents with the switch open...

 

[Spudnik]

Double posting because I figured it out:

I₁= 72/83 A
I₂= -51/83 A
I₃= -123/83 A

Using
I₂=I₁+I₃
12+4I₃-7I₁=0
12-9-5I₂-7I₁=0

Setting them up for a matrix:

I₁ + I₂ + I₃ = Voltage
+1 -1 +1 =0
-7 +0 +4 =-12
-7 -5 +0 =-3

Letting the calculator do the rest for you: priceless.

 

[rwisz]

Hello,

I would like to provide some guidance on solving this problem using Kirchhoff's rules. Firstly, I would recommend starting by labeling the circuit with all known values and directions of current flow. This will help you visualize the circuit and understand the relationships between the different components.

Next, I would recommend applying Kirchhoff's first rule (Kirchhoff's current law) at each junction in the circuit. This will help you determine the currents at each junction, including the unknown currents I_1, I_2, and I_3. Remember that the sum of currents entering a junction must equal the sum of currents leaving the junction.

After applying Kirchhoff's first rule, you can then apply Kirchhoff's second rule (Kirchhoff's voltage law) to loops in the circuit. This will help you set up equations that relate the voltages and resistances in each loop. Remember that the sum of voltage drops must equal the sum of voltage rises in a closed loop.

By solving the equations obtained from Kirchhoff's first and second rules, you should be able to determine the values of I_1, I_2, and I_3. I would recommend checking your solution by applying Kirchhoff's first rule again to make sure the currents at each junction are consistent with your calculated values.

In summary, solving circuits with multiple emf's using Kirchhoff's rules involves careful labeling, application of Kirchhoff's laws, and solving equations to determine the unknown currents. I hope this helps guide you in solving this problem.