How do I solve the current of this resistor using Maxwell's Current Theorem?

Question

I've been trying to solve this using the method the prof. taught us, and I happen to know the answer but I can't reach it no matter how many times I've tried. The circuit in question is below:

I am asked to use Maxwell's circulating current theorem to find out the current at the $20  \Omega$ resistor. My method was to write out all the loop equations:

$20 = 15I_1 + 10I_2$

$10 = 15I_1 + 25I_2 + 15I_3$

$10 = 15I_2 + 35I_3$

then solve by method of elimination. My answers are as follows:

$I_1 = 2.44A$

$I_2 = -1.66A$

$I_3 = -0.43A$

The answer for $I_3$ is $-0.57$.

Am I on the right path? If not, can someone point out where I am going wrong and why? Thanks

Answer 1

As I understand the circulating current theorem, you're really missing the simplicity of the method. You don't assign each resistor its own current. Call the current through the small, rectangular loop on the far left $I_\ell$ ($\ell$ for left). Let that current be positive when it runs clockwise. Let the current in the central loop be $I_c$, and for the right loop be $I_r$. The loop equation for left-hand loop is then

$$20 = 5I_\ell + 10(I_\ell - I_c)$$

The current through the 10-ohm resistor is a combination of the two loops' currents, as this is where they make contact. As I said, the currents are not labeled by resistors, but rather by loops. This makes each loop more complicated, but it saves you one extra equation (otherwise, you would've needed four currents for four resistors).

See if you can find the equations for the other two loops. The central one will likely be trickiest. Then try to express the current in the resistor you're interested in as a combination of these three currents (you already have to do this anyway to set up the system of equations).

If this isn't the basic idea behind how you learned the circulating current approach, then I apologize.