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## Homework Statement

For the multi-loop circuit as shown, assuming the capacitor is initially uncharged, find the currents in each of the resistors at t=0 after the switches are closed.

So I believe there are 8 unknowns really because of the 8th branch with the capacitor. If you look at the attached diagram 1, You can see what I mean. The problem asks for the currents across each resistor so I will use I

_{1-7}for the currents and R

_{1-7}for the resistances. I

_{cap}is the current across the extra branch.

## Homework Equations

ƩV=0 loop rule

ƩI=0 junction rule

V=IR

## The Attempt at a Solution

Its a long process but I chose the directions I assumed the currents were moving in diagram 2.

The circuit has 4 junctions for which I have the following 4 equations.

I

_{1}=I

_{3}+I

_{4}

I

_{3}=I

_{2}+I

_{5}

I

_{cap}=I

_{4}+I

_{6}

I

_{7}=I

_{cap}+I

_{5}

Next, I used the loop rule. There are 6 possible loops, I wrote 4 equations, one for each of the three inner loops and one loop for the outside. Since capacitor is uncharged there is no resistance and current goes freely through it (correct?).

For the top loop, I took the current to be traveling counterclockwise and got:

ε

_{4}-I

_{1}R

_{1}-I

_{3}R

_{3}-I

_{2}R

_{2}=0

Middle loop (taken clockwise)

-I

_{4}R

_{4}+I

_{5}R

_{5}+I

_{3}R

_{3}=0

Lower loop (clockwise)

ε

_{3}-I

_{6}R

_{6}-I

_{7}R

_{7}=0

Finally I went clockwise around the outside of the circuit

ε

_{3}-I

_{6}R

_{6}+I

_{4}R

_{4}+I

_{1}R

_{1}-ε

_{4}+I

_{2}R

_{2}-I

_{5}R

_{5}-I

_{7}R

_{7}=0

I haven't even found a single example approaching this kind of complexity. I'm trying to keep from curling up in the fetal position and crying.

I am not sure about the directions of my currents, so I'm worried that there are problems in my equations above. Is this possible? I've been told it just comes out negative in the end. Its asking a lot I know, but can anybody tell if all these equations are right?

I haven't taken linear algebra, but I've been trying to make it into a matrix and I have no idea what I am doing there either. It seems pretty insurmountable with algebraic methods, but that's not really what I'm focused on right meow. I think I just want to know that what I'm doing so far is right, then I can worry about getting solutions to I

_{1-7}. You are absolute lifesavers. Thank you.

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