Hello OtherWindow,
There are actually a variety of methods to solve the problem in terms of Kirchhoff's circuit laws. Using different methods leads to different equations. However, they will all ultimately come up with same answer in the end (if done correctly and with self-consistency).
In my attempt to help below, I will try to stick as closely as possible to the methods in which you have already started. Just keep in mind, the hints that I describe are not the only way to solve the problem.
OtherWindow said:
Homework Statement
I am using these variables:
L for the square length - so the rectangle on the left is L/2 by L
B' = rate at which field increases
λ = Resistance/Length
I
1 = Current on the right (so the top/right/bottom of square) - goes counterclockwise
I
2 = Current on the left (so the far left and the top/bottom on the little rectangle piece) - counterclockwise
I
3 = current through the middle wire (what the question is asking for) - unknown direction, I assumed downwards
I will stick with the above definitions.
(And this is what I meant by my first paragraph. You could have defined the current loops differently, but the final answer would still come out the same. That said, the above definitions are fine the way they are, so I will use those.)
Homework Equations
V = IR
Induced emf = derivative of flux
flux = ∫ B dA
The Attempt at a Solution
I have tried setting up loop-rule equations and then just using matrices to calculate the values of I.
Loop rule for the square:
V = B' * L2 = I1 * 3Lλ + I3 * Lλ
Okay, that's a valid way to approach the problem so far.
Loop rule for the whole outer rectangle
V = B' * 1.5 * L2 = I1 * 3Lλ + I2 * 2Lλ
The above equation is inconsistent with the rules that you outlined above. You defined a loop on the left, and a loop on the right. Never did you define I
1, or I
2, or any other I
n in terms of going around the whole outer rectangle. The above equation doesn't belong according to your own definitions of I
1, I
2 and I
3.
(By the way, you
could have defined your loops such that one of them is defined around the whole outer rectangle. Doing so means that you would be able to get rid of either one of your I
1 or I
2, since you wouldn't need both of them anymore.)
(As a general guideline, once all the components in a circuit have at least one current loop associated with it,
stop there! Defining more loops than are necessary doesn't help.)
Loop rule for the left rectangle:
V = B' * .5 * L2 = I2 * 2Lλ + I3 * Lλ
Which direction is I
3 pointed? Are you summing in the
same direction as I
3 or the
opposite? That has an impact on the
sign of the I
3 * Lλ term. Don't always assume everything is positive. If you're going against the direction of your own definition of a given current, the sign becomes negative.
Using the sum of currents in = sum of currents out
I1 = I2 + I3
Okay that looks fine.
So I got 4 equations. I remember from non-magnetic field type circuit analysis questions with 3 unknown currents I had to use 2 voltage equations + sum of current equations, because the 3 voltage equations would not give enough information to solve the system. My guess is that any 2 voltage equations and the current equation would be all I need.
Only 3 of your 4 equations are linearly independent (well, that and one has a minor mistake and the other doesn't even apply given the rules that were set up). You you really only have 3 independent equations.
I am getting the wrong answer with this set-up. Is there an error with my equations, or is this the completely wrong approach?
From here I suggest using substitution, linear algebra, or whatever your favorite method is for solving simultaneous equations.